APOS Theory
Ilana Arnon · Jim Cottrill · Ed Dubinsky Asuman Oktaç · Solange Roa FuentesMaría Trigueros · Kirk Weller
A Framework for Research and Curriculum Development in Mathematics Education
APOS Theory
Ilana Arnon • Jim Cottrill • Ed DubinskyAsuman Oktac • Solange Roa FuentesMarıa Trigueros • Kirk Weller
APOS Theory
A Framework for Research and CurriculumDevelopment in Mathematics Education
Ilana ArnonGivat Washington AcademicCollege of EducationTel Aviv, Israel
Jim CottrillDepartment of MathematicsOhio Dominican UniversityColumbus, Ohio, USA
Ed DubinskySchool of EducationUniversity of MiamiMiami, Florida, USA
Asuman OktacDepartamento de Matematica EducativaCinvestav-IPNMexico City, Mexico
Solange Roa FuentesEscuela de MatematicasUniversidad Industrial de SantanderBucaramanga, Colombia
Marıa TriguerosDepartamento de MatematicasInstituto Tecnologico Autonomo de MexicoCol. Tizapan, San Angel, Mexico
Kirk WellerDepartment of MathematicsFerris State UniversityBig Rapids, Michigan, USA
ISBN 978-1-4614-7965-9 ISBN 978-1-4614-7966-6 (eBook)DOI 10.1007/978-1-4614-7966-6Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2013942393
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Preface
The interest in producing this book arose out of a concern for the education of
graduate students and young researchers in mathematics education. A few years
ago, as a result of her experience in giving seminars and workshops about APOS
Theory, Marıa Trigueros, a member of the writing team for this book, raised the
issue in a conversation that took place among some of the authors. The conversation
centered around the question of why novice researchers find it so difficult to learn
and then to apply APOS Theory. In her view, published articles were not suffi-
ciently didactical to teach the theory in an effective manner. She believed a solution
could be offered by writing a book on APOS. Since then there has been consider-
able reflection about this project among the members of the writing team. This led
to a detailed plan that was followed by a long process of writing and revision and
finally to the book itself through a contract with Springer.
The fact that Ed Dubinsky, the founder of APOS Theory, as well as RUMEC1
members including two of Dubinsky’s former students, who learned the theory
from him through working on research projects together, and Solange Roa-Fuentes,
a new generation researcher who just finished her doctorate under the supervision of
one of the authors, all participated in the writing of this book, makes it unique. Like
the theory itself, people who study and apply APOS Theory are in constant
interaction, teaching this framework, learning from it and from each other.
The purpose of the book is to present a “portrait” of APOS Theory: to give a
detailed explanation of the theory, the basic principles behind it and its various
components; to describe the way in which research to develop the theory has been,
and continues to be, taking place; to show how it can be, and has been, used in
teaching; and to point to studies that report on the effectiveness of APOS-based
instruction. In the spirit of one pedagogical strategy most often used in connection
with APOS Theory, the authors of this book have worked cooperatively, with each
author intimately involved in writing, reviewing and revising every chapter. The
authors engaged in a joint process of discussion, writing, revision, discovery, and
1 The meaning of this acronym will be clarified in Chap. 2.
v
sometimes even debate, as they interacted with the theory in finding ways to explain
and present it in a single volume that encompasses more than 25 years of research
on student thinking and teaching. It is hoped that this cooperative work has led to a
comprehensive unity in the text that will be helpful to anyone interested in under-
standing and using APOS Theory: students, researchers, teachers, educators and
decision makers.
We thank everybody, especially the graduate students who constantly
challenged us through their questions. In particular we thank Yanet Gonzalez,
a graduate student at Cinvestav, for providing some of the information for the
annotated bibliography. We would also like to acknowledge Annie Selden who read
parts of the draft of this book and made many helpful suggestions.
vi Preface
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 From Piaget’s Theory to APOS Theory: Reflective
Abstraction in Learning Mathematics and the Historical
Development of APOS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1 Piaget’s Notion of Reflective Abstraction . . . . . . . . . . . . . . . . . . 6
2.2 Reflective Abstraction and the Antecedents
of APOS Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 First Thoughts About APOS Theory, 1983–1984 . . . . . . . . . . . . 10
2.4 First Developments of APOS Theory, 1985–1995 . . . . . . . . . . . 11
2.5 RUMEC, 1995–2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.6 Beyond RUMEC, 2003–Present . . . . . . . . . . . . . . . . . . . . . . . . 14
2.7 Related Theoretical Perspectives . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Mental Structures and Mechanisms: APOS Theory
and the Construction of Mathematical Knowledge . . . . . . . . . . . . . 17
3.1 Preliminary Aspects and Terminology . . . . . . . . . . . . . . . . . . . . 17
3.2 Description of Mental Structures and Mechanisms . . . . . . . . . . . 18
3.2.1 Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.2 Interiorization and Processes . . . . . . . . . . . . . . . . . . . . . 20
3.2.3 Encapsulation and Objects . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.4 De-encapsulation, Coordination, and Reversal
of Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2.5 Thematization and Schemas . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Overview of Structures and Mechanisms . . . . . . . . . . . . . . . . . . 25
4 Genetic Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1 What Is a Genetic Decomposition? . . . . . . . . . . . . . . . . . . . . . . 27
4.1.1 A Genetic Decomposition for Function
(Based on Ideas from Dubinsky 1991) . . . . . . . . . . . . . . 29
4.1.2 A Genetic Decomposition for Induction
(Dubinsky 1991, pp. 109–111) . . . . . . . . . . . . . . . . . . . . 30
vii
4.2 The Design of a Genetic Decomposition . . . . . . . . . . . . . . . . . . 33
4.2.1 Genetic Decomposition for Spanning Set and Span
(Based on Ku et al. 2011) . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2.2 Prerequisite Constructions . . . . . . . . . . . . . . . . . . . . . . . 36
4.2.3 Mental Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Role of the Genetic Decomposition in Research . . . . . . . . . . . . . 37
4.4 A Genetic Decomposition Is Not Unique . . . . . . . . . . . . . . . . . . 40
4.4.1 Prerequisites for the Construction of the Linear
Transformation Concept . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.2 Genetic Decomposition 1 . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4.3 Genetic Decomposition 2 . . . . . . . . . . . . . . . . . . . . . . . . 42
4.4.4 Genetic Decompositions 1 and 2: Constructing Process
and Object Conceptions of Linear Transformation . . . . . . 44
4.5 Refinement of a Genetic Decomposition . . . . . . . . . . . . . . . . . . 44
4.6 Role of the Genetic Decomposition in the Design
of Teaching Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.6.1 Genetic Decomposition of a Vector Space . . . . . . . . . . . 48
4.6.2 Activities Designed to Facilitate Development
of the Vector Space Schema . . . . . . . . . . . . . . . . . . . . . . 49
4.7 What Is Not a Genetic Decomposition . . . . . . . . . . . . . . . . . . . . 51
5 The Teaching of Mathematics Using APOS Theory . . . . . . . . . . . . 57
5.1 The ACE Teaching Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2 ISETL: A Mathematical Programming Language . . . . . . . . . . . . 59
5.2.1 A Brief Introduction to ISETL . . . . . . . . . . . . . . . . . . . . 59
5.2.2 The Syntax Is Close to Standard
Mathematical Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2.3 Supporting Mathematical Features . . . . . . . . . . . . . . . . . 62
5.2.4 Operations on Data Types . . . . . . . . . . . . . . . . . . . . . . . 64
5.2.5 ISETL as a Pedagogical Tool . . . . . . . . . . . . . . . . . . . . . 65
5.3 Teaching and Learning Groups . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.3.1 Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.2 Class Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.4 Results of the Instruction . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4 Application of the ACE Teaching Cycle in a Unit
on Repeating Decimals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4.1 First Iteration of the Cycle . . . . . . . . . . . . . . . . . . . . . . . 78
5.4.2 Second Iteration of the Cycle . . . . . . . . . . . . . . . . . . . . . 82
5.4.3 Third Iteration of the Cycle . . . . . . . . . . . . . . . . . . . . . . 85
5.4.4 Results of the Instruction . . . . . . . . . . . . . . . . . . . . . . . . 90
5.5 Analysis of Instruction Using the Research Framework . . . . . . . 91
viii Contents
6 The APOS Paradigm for Research and Curriculum
Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.1 Research and Curriculum Development Cycle . . . . . . . . . . . . . . 93
6.2 Data Collection and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.1 Interviews . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6.2.2 Written Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
6.2.3 Classroom Observations . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2.4 Textbook Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2.5 Historical/Epistemological Analysis . . . . . . . . . . . . . . . . 103
6.3 Types of APOS-Based Research Studies . . . . . . . . . . . . . . . . . . 104
6.3.1 Comparative Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.2 Non-comparative Studies . . . . . . . . . . . . . . . . . . . . . . . . 105
6.3.3 Studies of the Level of Cognitive Development . . . . . . . . 106
6.3.4 Comparisons of Student Attitudes and the Long-Term
Impact of APOS-Based Instruction . . . . . . . . . . . . . . . . . 107
6.4 Scope and Limitations of APOS-Based Research . . . . . . . . . . . . 107
7 Schemas, Their Development and Interaction . . . . . . . . . . . . . . . . . 109
7.1 Schemas in Piaget’s Work and in APOS Theory . . . . . . . . . . . . . 109
7.2 Examples of Schemas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.3 Development of a Schema in the Mind of an Individual . . . . . . . 112
7.4 Examples of Development of a Schema . . . . . . . . . . . . . . . . . . . 114
7.4.1 The Intra-Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.4.2 The Inter-stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.4.3 The Trans-stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.5 Assimilation of New Constructions into a Schema . . . . . . . . . . . 122
7.6 Interaction of Schemas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.6.1 Two Studies of Students’ Calculus Graphing Schema . . . 123
7.6.2 The Development of the Calculus Graphing Schema . . . . 124
7.7 Thematization of a Schema . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
Appendix: Problems for the Interview in the Chain Rule Study
(Cottrill 1999) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
8 Totality as a Possible New Stage and Levels in APOS Theory . . . . 137
8.1 Progression Between Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2 Stages and Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8.2.1 Piaget’s Work on Stages and Levels . . . . . . . . . . . . . . . . 139
8.2.2 Levels in APOS Theory . . . . . . . . . . . . . . . . . . . . . . . . . 139
8.3 A New Stage in the Infinity Studies . . . . . . . . . . . . . . . . . . . . . . 140
8.3.1 The Introduction of a New Stage . . . . . . . . . . . . . . . . . . 141
8.4 Levels Between Stages in 0:�9 . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4.1 Action to Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4.2 Process to Totality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8.4.3 Totality to Object . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.5 Previous Uses of the Idea of Totality . . . . . . . . . . . . . . . . . . . . . 148
8.6 The Tentative Nature of Totality as a Stage . . . . . . . . . . . . . . . . 149
Contents ix
9 Use of APOS Theory to Teach Mathematics
at Elementary School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
9.1 Applying APOS Theory in Elementary School Versus
Applying It in Postsecondary School . . . . . . . . . . . . . . . . . . . . . 152
9.2 Comparing a Standard Instructional Sequence
to an Instructional Sequence Based on APOS Ideas . . . . . . . . . . 154
9.3 Levels and Genetic Decompositions for the Transition
from Action to Process of Some Fraction Concepts . . . . . . . . . . 161
9.3.1 Levels in the Developments of Some
Fraction Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.3.2 Genetic Decompositions for the Concepts
That Were Investigated . . . . . . . . . . . . . . . . . . . . . . . . . 163
9.3.3 Additional Achievements: Abstract Objects . . . . . . . . . . 163
9.4 Manipulating Concrete Objects in the Imagination . . . . . . . . . . . 164
9.4.1 Criterion 1: The Student Declared Explicitly
That the Answer He or She Had Provided
Was a Result of Actions Which He or She Had
Performed on Imaginary Concrete Objects . . . . . . . . . . . 165
9.4.2 Criterion 2: Activating Imaginary Circle
Cutouts That Did Not Exist in the Original
Set of Manipulatives . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
9.4.3 Criterion 3: The Use of Drawings . . . . . . . . . . . . . . . . . . 167
9.4.4 Criterion 4: Verbal Indications That Involve
the Use of Language That Refers to the Concrete
Manipulatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
9.4.5 Criterion 5: Gestural Indications . . . . . . . . . . . . . . . . . . . 168
9.4.6 Criterion 6: Prompting . . . . . . . . . . . . . . . . . . . . . . . . . . 168
9.5 Equivalence Classes of Fractions in Grade 5 . . . . . . . . . . . . . . . 169
9.5.1 Equivalence Classes of Fractions in the Literature . . . . . . 170
9.5.2 The Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
9.6 What Is Known About the Use of APOS Theory
in Elementary School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Appendix: Fractions as Equivalence Classes: Definition . . . . . . . . . . . 174
10 Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.1 Questions About Structures, Mechanisms,
and the Relationship between APOS Theory
and the Work of J. Piaget . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
10.2 Questions Related to Genetic Decomposition . . . . . . . . . . . . . . 177
10.3 Questions About Instruction and Performance . . . . . . . . . . . . . 178
10.4 Questions Related to Topics Discussed in Mathematics
Education: Representations, Epistemology, Metacognition,
Metaphors, Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
10.5 A Question About Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
10.6 Questions About How Specific Concepts Can Be
Approached with APOS Theory . . . . . . . . . . . . . . . . . . . . . . . 185
x Contents
11 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
11.1 Developmental vs. Evaluative Nature . . . . . . . . . . . . . . . . . . . . 189
11.2 Macro-Level Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
11.3 A View of the Future of APOS . . . . . . . . . . . . . . . . . . . . . . . . 191
11.4 APOS Theory at a Glance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
11.4.1 Structures and Mechanisms . . . . . . . . . . . . . . . . . . . . 192
11.4.2 Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . 193
11.4.3 Pedagogical Approach . . . . . . . . . . . . . . . . . . . . . . . . 193
11.4.4 An Integrated Theory . . . . . . . . . . . . . . . . . . . . . . . . . 194
11.5 Last Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
12 Annotated Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.1 A Through B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
12.2 C Through De . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
12.3 Dubinsky (as Lead Author) . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.4 E Through M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
12.5 Works of Piaget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
12.6 P Through T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
12.7 V Through Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
About the Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
Contents xi
Chapter 1
Introduction
The acronym APOS stands for Action, Process, Object, and Schema. APOS Theory
is a theory of how mathematical concepts can be learned. Rooted in the work of
Jean Piaget, its fundamental ideas were first introduced in the early 1980s
(Dubinsky 1984), and since that time, extensive development and application
have been carried out by researchers, curriculum developers, and teachers in
many countries throughout the world.
APOS Theory focuses on models of what might be going on in the mind of an
individual when he or she is trying to learn a mathematical concept and uses these
models to design instructional materials and/or to evaluate student successes and
failures in dealing with mathematical problem situations. APOS Theory can be
used, and in many studies has been used, successfully, as a strictly developmental
perspective (e.g., Breidenbach et al. 1992), as a strictly analytical evaluative tool
(e.g., Dubinsky et al. 2013), or as both (e.g., Weller et al. 2011). APOS-based
research and curriculum development has focused mainly on learning mathematics
by students in the secondary and postsecondary grades, but as will be seen in
Chap. 9, work has also been done in the context of elementary and middle school
mathematics. There is also some preliminary work on applying APOS Theory to
areas outside of mathematics, such as computer science.
APOS is a constructivist theory. In Chap. 2 the sense in which that statement is
made will be explained. Chapter 2 also contains a description of Piaget’s notion of
reflective abstraction and the role it has played in the development of APOS
Theory. The development of APOS Theory to date is described in terms of three
major periods: first thoughts, work done by the Research in Undergraduate Mathe-
matics Education Community (RUMEC), and continuing efforts by small teams
that function independently.
Chapter 3 discusses the mental structures that constitute APOS Theory: Action,
Process, Object, and Schema and some of the mechanisms by which those mental
structures are constructed—interiorization, encapsulation, coordination, reversal,
and de-encapsulation. The discussion in this chapter is exemplified by several
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_1,© Springer Science+Business Media New York 2014
1
specific mathematical concepts that illustrate how APOS Theory can be used to
hypothesize the construction of mathematical knowledge by an individual. There is
also a discussion of some general issues that should be taken into consideration
when working with APOS Theory.
Chapter 4 presents an in-depth description of one of the major tools used in
APOS-based research and curriculum development—the genetic decomposition—
a hypothetical model of mental constructions that a student may need to make
in order to learn a mathematical concept. The chapter includes discussion of what
a genetic decomposition is and its role in working with APOS Theory. Several
issues regarding genetic decompositions, such as their nonuniqueness, the
relation between a preliminary genetic decomposition and its refinement(s), and
common misunderstandings about the design of a genetic decomposition, are
also considered.
Chapter 5 is concerned with the design and implementation of instruction using
APOS Theory. Implementation is usually carried out using the Activities, Class
discussions, Exercises (ACE) Teaching Cycle, an instructional approach that supports
development of the mental constructions called for by a genetic decomposition.
The ACE cycle includes activities on which students typically work coopera-
tively using a mathematical programming language such as ISETL (Interactive
SET Language). All of the components of the ACE Teaching Cycle and some
features of ISETL are described and examples are given of APOS-based instruction
on groups in abstract algebra for mathematics majors and on infinite repeating
decimals in a course for prospective elementary and middle school teachers.
In Chap. 6, the overarching research stance linked to APOS Theory is presented
as a paradigm, which differs from most mathematics education research trends in its
theoretical approach, methodology, and types of results. Following the ideas of
Kuhn (1962), this paradigm contains theoretical, methodological, and pedagogical
components that are closely linked together. It is pointed out that not all studies that
adopt APOS as a theoretical framework make use of all the elements of the
paradigm referred to in this chapter. Rather, it serves as an “ideal” organization
of an APOS-based research study.
Schemas and the thematization of a Schema are the central topics of Chap. 7.
The chapter begins with a general description of a schema and several examples.
Then, there is a description of how Schemas may develop and of the consequences
that result from modifications of a Schema, either through the introduction of new
information or through the interaction of one Schema with another. The chapter
ends by dealing with the issue of thematization, the mechanism involved in
constructing an Object conception of a Schema.
Chapter 8 contains three themes: a general discussion of the progression between
stages in APOS Theory and related pedagogical strategies, a description of the
terms stages and levels as they appear in the work of Piaget and in APOS-based
research, and a summary of the research regarding 0:�9 that suggested the need for
2 1 Introduction
levels between stages and for Totality as a new stage between Process and Object.1
At this point, the status of Totality and the use of levels described here are no more
than tentative because evidence for a separate stage and/or the need for levels arose
out of just two investigations: the studies of the relation between 0:�9 and 1, which isdiscussed in Chap. 8, and the study of fractions, considered in Chap. 9. Thus, it
remains for future research to determine if Totality can be considered as a separate
stage, if levels are really needed in these contexts, and to explore what the mental
mechanism(s) for constructing them might be. Research is also needed to determine
the role of Totality and levels in other contexts, both those involving infinite
processes and those involving finite processes.
Chapter 9 discusses a 1990s project that involved the introduction of Piagetian and
APOS ideas into the teaching of fractions in grades 4 and 5. The chapter describes the
data and conclusions from three studies related to that project (Arnon 1998; Arnon
et al. 1999, 2001). In contrast to most APOS-based studies, these investigations
describe the possibility of applying APOS Theory to investigate the learning of
mathematics in elementary school. In elementary and middle school grades, most
students are at what Piaget defined as the stage of concrete operations, which means
that the objects acted upon by actions must be concrete (e.g., blocks, cutouts,
drawings), that is, they need to be perceivable by one’s senses (Piaget 1975, 1974/
1976). At the higher grades, however, the objects on which actions are performed are
not necessarily concrete objects that belongmainly to the physicalworld but rather can
be abstract objects (e.g., propositions, functions) that exist mainly in the minds of
individuals. Thus, from the perspective of APOS Theory, the concreteness of the
objects to which actions are applied constitutes the main difference between the
elementary or middle grades and the secondary or postsecondary grades. In addition,
the chapter describes how imagination plays a substantial role in a young child’s
interiorization of an Action into a Process. Another study (Arnon et al. 2001),
discussed in Chap. 9, describes how using concrete representations following some
APOSprinciples enabled grade 5 students to learn abstractmathematical concepts that
are outside the regular syllabi of elementary school curricula.
Chapter 10 gives some questions that have been asked about APOS Theory
either in print or in personal communication with the authors of this book and
provides suggested responses to these questions. The topics range among specific
components of the theory, their relationship to the work of Jean Piaget, and dealing
with particular concepts in mathematics courses. Other topics from mathematics
education research, such as context, epistemology, intuition, metacognition,
metaphors, and representations, are discussed.
Chapter 11 summarizes the entire book by exploring themes and common
threads. The notions of the developmental/evaluative dichotomy, the growth of
the theory through mechanisms similar to those used in the theory, and the future of
1 Per convention, the bar over the digits in a decimal expression signifies the digits that repeat in a
repeating decimal.
1 Introduction 3
APOS Theory are discussed. The chapter provides “APOS Theory at a glance” and
ends with some final thoughts.
Finally, Chap. 12 contains an extensive annotated bibliography of publications
related to APOS Theory.
Although the intention has been to make this book as up to date as possible,
whatever success has been achieved in that endeavor may not last. APOS Theory is
a living, growing body of ideas that attempts to synthesize the thinking of its
progenitor, Piaget, with that of current and future workers, along with data resulting
from empirical studies of students trying to learn mathematics. The result is that
those involved in APOS-based research and curriculum development are continu-
ally revising and rethinking various aspects of the theory and making revisions
where appropriate. This is in keeping with Piaget who once wrote that “‘Piaget’s
Theory’ is not completed at this date and the author of these pages has always
considered himself one of the chief revisionists of Piaget” (Piaget 1975, p. 164).
This means that this book cannot, and should not, be considered as the “last word”
on APOS Theory. Future studies will lead to further revisions. Indeed, the reader
can see in this text that this has already occurred. Although almost all of the work
done in APOS Theory during the first 30 years of its life is still relevant, the book
points out several examples in which thinking about one or another aspect of the
theory has changed and, accordingly, various descriptions have been modified.
In some cases, material about APOS Theory that appears in published works is
used (with attribution), but revisions are made to reflect current thinking and the
results of APOS-based research. There is even one example (see the discussion of
Totality in Chap. 8) where a major change in the theory has been proposed. It is
hoped that this dynamic nature of APOS Theory makes it even more helpful to
researchers, teachers, and students.
4 1 Introduction
Chapter 2
From Piaget’s Theory to APOS Theory:
Reflective Abstraction in Learning
Mathematics and the Historical
Development of APOS Theory
The aim of this chapter is to explain where APOS Theory came from and when it
originated. A discussion of the main components of APOS Theory—the mental
stages or structures of Action, Process, Object, and Schema and the mental
mechanisms of interiorization, coordination, reversal, encapsulation, and
thematization—points to when they first came on the scene and how their meanings
developed. The published research of those involved in the development of APOS
Theory, which includes some early colleagues and students of Dubinsky as well as
those who were members of the Research in Undergraduate Mathematics Education
Community (RUMEC), is described. The descriptions in this chapter are very brief
and will be expanded in later chapters.
In the seven sections of this chapter, the development of APOS Theory and its
application in helping students construct their understanding of various mathemati-
cal concepts is traced. This development, which began in the early 1980s, has
continued since that time and plays an important role in mathematics education
research and curriculum development. The chapter begins with a description of
Piaget’s notion of reflective abstraction (Sect. 2.1) and how it inspired the develop-
ment of APOS Theory (Sect. 2.2). Next there are descriptions of the first thoughts
about APOS Theory (Sect. 2.3) and its first major period of development (Sect. 2.4).
The second major period began with the formation of RUMEC, which produced a
large amount of coordinated cooperative research conducted by teams of
mathematicians who were moving into education research (Sect. 2.5). During the
third major period, from the end of RUMEC as a formal organization to the present,
various teams have been functioning independently to conduct APOS-based
research and to study its application to the design and implementation of instruction
(Sect. 2.6). Finally, there is a very brief mention of two related theoretical
perspectives (Sect. 2.7).
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_2,© Springer Science+Business Media New York 2014
5
2.1 Piaget’s Notion of Reflective Abstraction
One of the major ideas of Piaget is what he called reflective abstraction, which he sawas both the main mechanism for the mental constructions in the development of
thought and the mental mechanism by which all logico-mathematical structures are
developed in the mind of an individual. He reiterated this point of view very often in
many different contexts. For example, regarding the development of thought, he
wrote, “The development of cognitive structures is due to reflective abstraction. . .”(Piaget 1975/1985, p. 143). Regarding mathematics, he considered that reflective
abstraction is the mental mechanism by which all logico-mathematical structures are
derived (Piaget 1967/1971) and wrote that “. . .it [reflective abstraction] alone
supports and animates the immense edifice of logico-mathematical construction”
(Piaget 1974/1980, p. 92). Starting around 1982, a researchmathematician, Dubinsky,
switched from research in mathematics (functional analysis) to research into the
mental activities involved in students’ learning of mathematics. It was statements
like the above about the edifice of mathematics that first attracted him to Piaget’s
ideas about reflective abstraction and their application to mathematical thinking.
What is reflective abstraction? Piaget’s answer, repeated in many different
publications, consists of two parts. The first part involves reflection, in the sense
of awareness and contemplative thought, about what Piaget called content and
operations on that content, and in the sense of reflecting content and operations
from a lower cognitive level or stage to a higher one (i.e., from processes to
objects). The second part consists of reconstruction and reorganization of the
content and operations on this higher stage that results in the operations themselves
becoming content to which new operations can be applied (Piaget 1973). This
second step appeared to Dubinsky to be very close to certain mathematical ideas.
One of many examples is the case of functions. They are first constructed as
operations that transform elements in a set, called the domain, into elements in a
set, called the range. Then, at a higher stage, as elements of a function space,
functions become content on which new operations are constructed. Integers are
another example. At one stage, an integer is an operation or process of forming units
(objects that are identical to each other) into a set, counting these objects and
ordering them. At a higher stage, integers become objects to which new operations,
e.g., those of arithmetic, are applied (Piaget 1965). These types of examples led
Dubinsky to believe that reflective abstraction could be a powerful tool in describ-
ing the mental development of more advanced mathematical concepts.
2.2 Reflective Abstraction and the Antecedents
of APOS Theory
To clarify the above ideas, it might be helpful to look at some of Piaget’s
examples of reflective abstraction and see how they formed the antecedents of
APOS Theory.
6 2 From Piaget’s Theory to APOS Theory. . .
Piaget did not believe that the most general and useful abstract ideas come from
drawing out common features of a variety of phenomena. Considering an example
from advanced mathematics, he wrote that:
[T]he group concept or property is obtained, not by this sort of abstraction [drawing
out common features], but by a mode of thought characteristic of modern mathematics
and logic—“reflective abstraction”—which does not derive properties from things,but from our ways of acting on things, the operations we perform on them. . .. (Piaget1968/1970, p. 19)
In other words, the development of knowledge about an object, either mental or
physical, requires both the object and a subject who acts on the object. In his view,
the subject (knower) and the object cannot be dissociated; it is impossible to speak
of either of them without the other. Piaget applied these ideas to the full range of
topics in mathematics, from the most elementary concepts constructed by the young
child to the advanced work of the research mathematician. This general
framework—content and operations on content that lead to the operations them-
selves becoming new content—lays the foundation for more subtle distinctions,
such as the distinction between material actions and interiorized operations, that
constitute the difference between the mental structures of Action and Process and
how mental mechanisms, such as interiorization and encapsulation, lead to the
formation of different conceptions that constitute the A!P!O!S progression.
Piaget made many statements such as:
. . . it follows that when the child discovers by experience the result of an action, for
example, that the result of an addition is independent of the order followed (which is a
property of the actions of combining and ordering and not a property of the objects as such,
which include neither sum nor order independently of the actions carried out on them),
reflective abstraction consists of translating a succession of material actions into a system of
interiorized operations, the laws of which are simultaneously implied in an act. (Beth and
Piaget 1966/1974)
What Piaget seems to be saying here is that properties of objects do not reside in
the objects but rather in the actions that are performed on these objects. Thus,
properties of objects depend on both the objects and on subjects who know the
objects. Dubinsky interpreted “material actions” in the above quote to refer to actions
that are performed by a subject but are external to the subject. In the example above,
the material actions consisted of transforming physical objects by taking two small
sets of objects; counting the number in one set, then the other set, and adding the two
results to get the total number of objects; and then repeating the action with the order
of the two sets reversed to see that the total is the same. Here, the objects are numbers
(integers represented by sets of physical objects), the action applied to these objects is
addition, and the property (of the operation, not the numbers) is commutativity. In
APOS Theory, Piaget’s “interiorized operations” became Processes.1 The “transla-
tion” became the APOS mental mechanism of “interiorization,” whereby an
1Capitalization is used to differentiate between Piaget’s terminology and Dubinsky’s use of
Action, Process, Object, and Schema.
2.2 Reflective Abstraction and the Antecedents of APOS Theory 7
external, that is, physical, Action (“material” action) is reconstructed in the mind of
the subject to become a Process (interiorized operation), that is, an internal, mental
construction that does the same thing as the Action, but wholly in the mind of the
subject rather than externally. Dubinsky considered Piaget’s “system” to refer to a
“Schema” which, in this case, is the concept of commutativity, and Piaget’s phrase
“which are simultaneously implied in an act” was the source of the idea of “coher-
ence,” by which a subject decides whether or not a particular Schema is applicable to
a particular mathematical problem situation.
Piaget’s notion of reflective abstraction also influenced the development, in
APOS Theory, of how a Process (interiorized operation or Action) is transformed
into an Object (operation to which new higher stage operations can be applied) via
the mental mechanism of encapsulation. Piaget applied reflective abstraction to the
concept of proportion when he wrote, “A proportion is an equality of relationships,
in other words, a specific case of equivalence between relations” (Piaget et al. 1968/
1977, p. 186). This is because a proportion begins with a relation between two
objects. For example, division of a positive integer a by a positive integer b, writtena
b, tells us that a certain number of copies of b is contained in a, which is a relation
between a and b. Then, there is the same relation between two other positive
integers c and d, writtenc
d. Each of these two relations is an action applied to the
pairs of positive integers a, b and c, d. Thus, the relations come from a learner’s
actions on these objects and so are examples of reflective abstraction. A further
reflective abstraction transforms these actions to fractions, which are objects to
which numerical values can be assigned.
Dubinsky’s interpretation of this description is that the relationsa
band
c
dare
Processes that are encapsulated to become Objects to which another relation,
comparison, can be applied. This latter relation is the meaning of a proportion,
which is itself an Action, not on physical objects, but on the relationsa
band
c
dwhich
become mental Objects as a result of applying the mental mechanism of
encapsulation.
Other examples of how APOS Theory arises out of Piaget’s studies of reflective
abstraction and the development of intelligence concern the notions of schema,
thematization of schemas, and coordination of schemas. Consider, again, the
concept of positive integer. Piaget (1965) describes a long period of development
of this concept, culminating around age 7.2 This development involves the con-
struction of a number of schemas, the main two being what Piaget called seriation
(ordering) and classification (formation and comparison of sets). The schema of
classification is the structure that allows the child to look at some objects (e.g.,
pieces of fruit) and think of them as indistinguishable units, ignoring all qualities. In
2As Piaget (1972) acknowledged, subsequent research showed that the age at which various
cognitive developments occur could vary as a function of the child’s culture and other factors
such as aptitudes and interests.
8 2 From Piaget’s Theory to APOS Theory. . .
applying this schema, the child uses the schema of 1–1 correspondence (previously
or simultaneously constructed) to determine that two such sets have the same
number of units or uses the set inclusion schema (also previously or simultaneously
constructed) to determine that one set has more (or less) elements than another. This
latter understanding lays the foundation for development of the schema of seriation
by which the child is able to imagine a sequence of sets such as
f1g; f1þ 1g; f1þ 1þ 1g; f1þ 1þ 1þ 1g; . . .3
With these constructions, the child can name these sets as one, two, three, four,
etc., and also name their position in the order as first, second, third, fourth, etc.
Finally, the two schemas of classification (set formation) and seriation (ordering)
are thematized and then coordinated to form a new schema. The key step in the
coordination occurs when the child realizes that, for example, the set with four
elements in the above sequence is also the fourth set in the sequence. The resulting
schema is the concept of positive integer. Piaget considered all of these
constructions to be examples of reflective abstraction.
The coordination referred to by Piaget is an action on two schemas. This is a very
general use of the term coordination that includes any construction which uses twoschemas, such as one schema following another or going back and forth between
the two schemas using parts of one and then of the other. In order to do this, the
schemas must first be thematized, which means made into objects (as processes are
encapsulated into objects) to which the action of coordination can then be applied.
The notion of an individual moving mentally from Action to Process, and from
Process to Object, arises clearly in Piaget’s discussion of the cognitive development
of functions, where he wrote, “Their most general characteristic stems from their
passage from qualitative coproperties resulting from elementary ‘applications’ to
operatorily quantifiable covariations, then to variations of variations, etc.” (Piaget
et al. 1968/1977, p. 186). Here, Piaget’s comment can be interpreted to refer to
functions as maps (application) which are initially actions and then processes
(operatorily quantifiable covariations) and then to objects (making it possible to
apply variations to variations.)
Dubinsky interpreted these types of passages as descriptions of cognitive devel-
opment that begins with Actions (elementary “applications”) that are interiorized
into Processes (operatorily quantifiable covariations) and then encapsulated into
Objects to which new Actions can be applied (variation of variations). This is an
example, rooted in Piaget’s reflective abstraction, of the development from Action
to Process to Object to Schema, the A!P!O!S progression that is the heart of
APOS Theory. Note, however, that although this progression is presented, of
necessity, as a linear string, the development does not always proceed linearly,
one stage after another. Rather, an individual may move back and forth between
stages as the situation requires.
3 Piaget is using the symbol “+” here, not as addition, but as “and.”
2.2 Reflective Abstraction and the Antecedents of APOS Theory 9
Thus this section has shown examples of how Piaget’s theory of reflective
abstraction formed the antecedents to APOS Theory—the mental structures of
Action, Process, Object, and Schema, and the mental mechanisms of interiorization,
coordination, reversal, encapsulation, and thematization as well as their formation
into the developmental A!P!O!S progression. This progression, together
with the APOS structures and some of the mechanisms, is illustrated in Fig. 2.1
which shows that Actions operate on Objects; Actions are interiorized into Pro-
cesses; Processes are encapsulated into Objects; and Objects are de-encapsulated
back to the Processes whence they came. The entire system is part of a Schema.
This diagram and variations of it will be used throughout the book.
2.3 First Thoughts About APOS Theory, 1983–1984
Dubinsky began, around 1983, to think about applying Piaget’s reflective abstraction
to postsecondary mathematics and to develop the ideas that later became APOS
Theory. The first publication concerning these ideas appeared in 1984 in the
proceedings of a conference in Helsinki, Finland, at which he was an invited speaker
(Dubinsky 1984). In this talk, he discussed the distinction between thinking about a
function as a Process and as an Object and spoke about using the experience of
computer programming to help students understand that distinction. He discussed
how one applies Actions to mental Objects, gave an example in which a Pascal
program represented a proof by mathematical induction, and expressed the view
(which he later investigated formally in Dubinsky 1986a, 1989) that if students
wrote, debugged, and used such a program, their development of an understanding
of induction would be enhanced.
At that time, Dubinsky was particularly interested in the use of computer
experiences to help students construct their understanding of mathematical
concepts. He wondered, for example, if working with computers in certain ways
could help students make reflective abstractions. He observed that the intensive
work in writing, debugging, and running computer programs tended to have a
profound effect on how the programmer thinks about the content of what he or
Fig. 2.1 APOS Theory (based on Asiala et al. 1996)
10 2 From Piaget’s Theory to APOS Theory. . .
she was representing on the computer. He thought that careful choices of computer
activities could change students’ thinking in ways that would help them learn
mathematics. Jack Schwartz, one of the few people elected to the US National
Academy of Sciences in both Mathematics and Computer Science and the creator of
the programming language SETL (forerunner of ISETL), once observed that the
only mathematical background necessary to learn to program in SETL was
contained in the content of a standard college first-year course in discrete mathe-
matics. Dubinsky noticed that there seemed to be a lot more people who were
successful in learning to program computers than in learning discrete mathematics.
So he decided to try to reverse Schwartz’ observation and have students learn to
program in SETL4 in order to learn topics in discrete mathematics. In the Helsinki
talk, he reported on a discrete mathematics course that he developed based on
that idea.
2.4 First Developments of APOS Theory, 1985–1995
During the period 1985–1988, Dubinsky, with various collaborators, developed
pedagogical methods for using programming to induce students to interiorize
Actions into Processes, encapsulate Processes into Objects, and apply the mental
structures that were constructed as a result of programming to learn various
mathematical concepts. This work led to the publication of the first textbook that
was based entirely on APOS Theory and the use of computer programming as a
pedagogical tool (Baxter et al. 1988). This was a textbook for a college course in
discrete mathematics that covered such topics as propositional and predicate calcu-
lus, sets and tuples, functions, combinatorics, matrices, determinants, mathematical
induction, relations, and graphs. The pedagogical strategy used throughout the text
was for the authors to develop, for each concept, a genetic decomposition, adescription of the mental structures of Actions, Processes, and Objects that students
might use in constructing an understanding of the concept and the mental
mechanisms (i.e., specific reflective abstractions) such as interiorization and encap-
sulation by which students might construct these structures. The text included
laboratory activities in which students used ISETL to write short computer
programs that were designed to help them to make the mental constructions called
for by the genetic decomposition. (See Chap. 4 for a more thorough discussion of
genetic decomposition.) The two most important programming activities involved
the interiorization of an Action to a Process (by writing a computer program that
performed the action on any appropriate input) and encapsulation of the Process to
an Object (by using the program as input and/or output to other programs).
Examples of these programming activities will be discussed in Chap. 5.
4 Later, SETL was replaced by the interpretive programming language, ISETL, developed by
G. Levin.
2.4 First Developments of APOS Theory, 1985–1995 11
During the period 1989–1995, Dubinsky continued working with various
collaborators to develop the framework that eventually became known as APOS
Theory. Following Piaget’s dictum that the nature of a concept (epistemology) is
inextricably interwoven with how it develops in the mind of an individual
(learning), all of the research into the epistemology of mathematics took place in
conjunction with one or more applications of the theory to teaching in actual
classrooms in various colleges. This preliminary APOS-based research and curric-
ulum development led to publications in refereed journals on such mathematical
concepts as mathematical induction, compactness, functions, predicate calculus,
and calculus. This research, together with the effectiveness of APOS-based peda-
gogical strategies, was reported in a number of conference presentations and
published papers that appeared during the period 1989–1997. These reports will
be described in some detail in Chap. 6. The pedagogical strategies eventually led to
the development of the ACE Teaching Cycle which is discussed in detail in Chap. 5.During the two periods, 1983–1984 and 1985–1995, the main components of
APOS Theory were introduced, developed, and understood pretty much in the way
they are understood today. These components include both the mental structures,
Actions, Processes, Objects, and Schemas, and the mental mechanisms for building
these structures, interiorization, coordination, reversal, encapsulation, and
thematization. At the Helsinki conference, in his first public report on this work,
Dubinsky began speaking about Actions, Processes, and Objects and, in particular,
about applying an Action to a Process (Dubinsky 1984). The term encapsulation, as
the mental mechanism for transforming a Process to a mental Object, was
introduced publicly about a year later (Dubinsky 1985). Shortly after, interioriza-
tion as the mechanism for converting an Action to a Process was first mentioned in
an APOS context (Dubinsky 1986b), although the idea of converting an external
Action to an internal Process was present from the beginning, as was the application
of an Action to a mental Object (Dubinsky 1984). Also discussed during that time
was the idea of transforming a Schema to an Object, which could be acted on by
another Schema (Dubinsky 1986b). At the time, this transformation was considered
to be an encapsulation, but later in Asiala et al. (1996), the name was changed to
thematization, the term presently in use, in order to maintain consistency with
Piaget (Piaget and Garcıa 1983/1989).
During this period APOS-related research did not pay much attention to the
mental structure of a Schema. While Schema was mentioned as a “more or less
coherent collection of objects along with actions which the subject can perform on
them” (Dubinsky 1986b, p. 2), nothing was said during this period about the
meaning of “coherent” Schema nor was any attempt made to distinguish the notion
of Schema from the “concept image” of Vinner and his colleagues (see Vinner
1983; Vinner and Dreyfus 1989). In fact, Schema and concept image differ in three
important ways. The first difference is illustrated in Vinner and Dreyfus (1989):
In most cases, he or she decides on the basis of a concept image, that is, the set of all themental pictures associated in the student’s mind with the concept name, together with all
the properties characterizing them. (By mental picture we mean any kind of
representation—picture, symbolic form, diagram, graph, etc.) The student’s image is a
12 2 From Piaget’s Theory to APOS Theory. . .
result of his or her experience with examples and nonexamples of the concept. Hence, the
set of mathematical objects considered by the student to be examples of the concept is not
necessarily the same as the set of mathematical objects determined by the definition.
(p. 356)
As can be seen from this quote, concept image is mainly concerned with the
mathematics involved in a concept whereas Schema describes the mental structures
involved in the mind of an individual who understands, or is developing an under-
standing of, that mathematics. The second difference is that a Schema can be
thematized to be an Object on which Actions can be performed and which can
become parts of other Schemas, whereas no such activities are discussed in the
literature on concept image. The third difference is the notion of coherence which
concerns the use of a Schema in dealing with mathematical problem situations and
which, again, is not considered in the literature on concept image. A more detailed
discussion of the nature of Schema, including the distinction between the stage of
Schema inAPOSTheory and concept image, will be given in Sect. 2.5 and inChap. 7.
Although Dubinsky spoke of a continuous development from Action to Process
to Object to Schema in many places during this period (see, e.g., Dubinsky 1991, for
a full discussion that is not very different from the current understanding of this
progression), the acronym APOS was not introduced until the next period (Cottrill
et al. 1996).
2.5 RUMEC, 1995–2003
During the period 1988–1996, Dubinsky was the recipient of grants from the US
National Science Foundation (NSF) to conduct curriculum development projects in
undergraduate mathematics courses. The research for the paper on reflective
abstraction (Dubinsky 1991) and several reports on calculus (not directly related
to APOS Theory) were supported by these grants. However, he had also collected a
huge amount of data, mostly from interviews with college students who had been
studying mathematical concepts in APOS-based courses in calculus, discrete math-
ematics, and abstract algebra. It was impossible for one person, or even a team of
two or three researchers, to analyze all of this data and report the results. So the data
remained unanalyzed at that time.
Then, in 1995, Dubinsky received a 5-year grant from the NSF to conduct
summer workshops on professional development for college mathematics faculty
to learn more about cooperative learning. The project was titled Cooperative
Learning in Undergraduate Mathematics Education (CLUME). In one of the work-
shop sessions during Summer, 1995, Dubinsky discussed the use of cooperative
learning in his curriculum development projects and mentioned in passing the
existence of this unanalyzed data. The participants expressed considerable interest
in hearing about the research that generated this data and so an “off-line” unofficial
evening session was held to talk about research in undergraduate mathematics
education.
2.5 RUMEC, 1995–2003 13
It turned out that most of the CLUME participants were mathematicians who
were interested in changing their research efforts from mathematics to undergradu-
ate mathematics education. So an organization, RUMEC, was formed. Its purpose
was to help mathematicians get started in doing education research. The initial
method, in the spirit of cooperative learning, was to conduct cooperative research in
small teams of three, four, or five researchers, by analyzing Dubinsky’s data
and reporting the results in papers submitted for publication. This work was funded
for 5 years, from 1996 until 2001, by two grants from the Exxon Educational
Foundation.
An important feature of the work of RUMEC was something that was unique to
education research in mathematics: each team conducted its research and produced
a draft of a research paper that was distributed by email to all of the approximately
25 members of the organization. The manuscript was reviewed by all RUMEC
members, with suggestions for revisions sent to the authors. It was then discussed
by the entire group at its annual meeting. This process was called internal review.It was only advisory as the authors made the final decisions on the suggestions,
produced a final manuscript, and submitted it for publication. The process
was apparently effective because during the period 1995–2003, RUMEC teams
submitted 14 manuscripts, and, although some journals requested revisions, every
RUMEC submission was eventually accepted for publication!
These publications dealt with the development of APOS Theory (Asiala et al.
1996; Clark et al. 1997; Dubinsky and McDonald 2001; Weller et al. 2003) and with
learning specific mathematical concepts including limits (Cottrill et al. 1996);
graphing and the derivative (Asiala et al. 1997a; Baker et al. 2000); the chain rule
(Clark et al. 1997); cosets, normality, and quotient groups (Asiala et al. 1997b);
binary operations, groups, and subgroups (Brown et al. 1997); permutations and
symmetries (Asiala et al. 1998); sequences and series (McDonald et al. 2000);
fractions (Arnon et al. 2001); and the definite integral (Czarnocha et al. 2001).
There was also one study on student attitudes about their experiences with peda-
gogy based on APOS Theory (Clark et al. 1999). Details of some of these studies
will be presented in subsequent chapters, but two publications during this period
deserve special mention. The first (Asiala et al. 1996) gives a complete and coherent
description of APOS Theory as it stood at that time; a description of the ACE
Teaching Cycle, which is the main pedagogical strategy for APOS-based instruc-
tion; and the methodology used in APOS-based research. The second (Weller et al.
2003) summarizes the results of APOS-based research and the ACE Teaching
Cycle up until that time.
2.6 Beyond RUMEC, 2003–Present
When the Exxon funding ran out in 2001, it was no longer possible to hold meetings
of the RUMEC membership. Internal reviews were still conducted for a while, but
by 2003, RUMEC no longer existed as an organization. Nevertheless, individual
14 2 From Piaget’s Theory to APOS Theory. . .
members of RUMEC continue to form research teams for individual studies. There
are two examples in which this “post-RUMEC” research has led to further
developments of APOS Theory.
The first development involves the structure of Schema. Although the idea of a
Schema becoming an Object that can be acted on by a Process or by another
Schema emerged very early in the development of APOS Theory (Dubinsky
1986b), this idea, now referred to as thematization of a schema, was not extensivelystudied until Cooley et al. (2007). This development and the interaction of schemas
will be discussed in full detail in Chap. 7.
The second development has to do with a series of studies that use APOS Theory
to investigate the development of students’ understanding of the mathematical
concept of infinity (Weller et al. 2004; Dubinsky et al. 2005a, b, 2008, 2013;
Stenger et al. 2008; Brown et al. 2010; Weller et al. 2009, 2011). These studies
led to the introduction of a potential new stage in APOS Theory, namely, Totality,
which lies between Process and Object. The terms “totality” and “total entity” had
been used since the beginning of APOS Theory (Dubinsky 1984), but it was always
considered to be just part of the Object stage. Totality as a separate stage and the
reasons for its introduction will be discussed in Chap. 8.
In Mexico a group of researchers are currently conducting an APOS-based
project to study mental constructions involved in learning linear algebra concepts
(such as vector space, basis, linear transformations, spanning sets, and systems of
linear equations) and to make pedagogical suggestions for courses on these topics
(see, e.g., Ku et al. 2008; Oktac and Trigueros 2010; Parraguez and Oktac 2010;
Roa-Fuentes and Oktac 2010; Trigueros and Oktac 2005).
2.7 Related Theoretical Perspectives
Shortly after Dubinsky began speaking about Actions, Processes, Objects, and
Schemas, Sfard began to speak about operational and structural conceptions
(Sfard 1987), which she later changed to process and object (Sfard 1991). Some-
what later, Gray and Tall introduced the notion of a procept, which is an amalgam
of three components: a process, an object that is produced by that process, and a
symbol which is used to represent either the process or the object (Gray and Tall
1994). Although both of these theoretical perspectives have some commonalities
with APOS Theory, there are important differences. For example, neither of them
deals with Actions or the construction of Processes (as in interiorizing Actions to
Processes). Also, there is no mention of anything like schemas, although Tall and
Vinner (1981) discuss concept image, which, as was indicated in Sect. 2.4, is very
similar to a Schema, but with some differences.
Finally, these two approaches do not put emphasis on the application of their
theoretical perspectives to the design and implementation of instruction.
2.7 Related Theoretical Perspectives 15
Chapter 3
Mental Structures and Mechanisms:
APOS Theory and the Construction
of Mathematical Knowledge
The focus of this chapter is a discussion of the characteristics of the mental
structures that constitute APOS Theory, Action, Process, Object, and Schema,
and the mechanisms, such as interiorization, encapsulation, coordination, reversal,
de-encapsulation, thematization, and generalization, by which those mental
structures are constructed.
In Sect. 3.1, general aspects of mental structures, mental mechanisms, and their
role in the development of an individual’s understanding of mathematical concepts
are considered. In Sect. 3.2, these mental constructions and mechanisms are
explained and examples are provided to illustrate how APOS Theory describes an
individual’s construction of mathematical knowledge. These ideas are summarized
in Sect. 3.3.
3.1 Preliminary Aspects and Terminology
APOS Theory is principally a model for describing how mathematical concepts can
be learned; it is a framework used to explain how individuals mentally construct
their understandings of mathematical concepts. From a cognitive perspective, a
particular mathematical concept is framed in terms of its genetic decomposition, adescription of how the concept may be constructed in an individual’s mind. This
differs from a mathematical formulation of the concept, which deals with how the
concept is situated in the mathematical landscape—its role as a mathematical idea.
Individuals make sense of mathematical concepts by building and using certain
mental structures (or constructions) which are considered in APOS Theory to be
stages in the learning of mathematical concepts (Piaget and Garcıa, 1983/1989).
These structures arise through instances of reflective abstraction (as discussed in
Chap. 2), which, in APOS theory, consists of mental mechanisms such as interiori-
zation, encapsulation, coordination, reversal, de-encapsulation, and thematization.
Since a genetic decomposition is hypothesized theoretically and tested empirically,
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_3,© Springer Science+Business Media New York 2014
17
it can serve as a powerful descriptive and predictive tool to describe an individual’s
mathematical thinking. By detailing the structures involved in learning a particular
concept, a genetic decomposition can help an instructor to uncover sources of
difficulty that arise in the learning process. By providing a description of how a
concept might develop in the mind of an individual, a genetic decomposition can
help to guide the design of instruction. Genetic decompositions are addressed in
considerably more detail in Chap. 4, and their role in instruction and research is
explained further in Chaps. 5 and 6.
In APOS-based research, the terms conception and concept appear quite fre-
quently. Although related, they are different ideas. McDonald et al. (2000) describe
the distinction as follows:
We distinguish between conception and concept as the first is intrapersonal (i.e., the
individual’s idea or understanding) and the latter is communal (i.e., a concept as agreed
upon by mathematicians). (p. 78)
For a particular piece of mathematical content, a conception develops as a
result of reflective activity. The term concept refers to the collective understand-
ing of that content by the community of mathematicians. Thus, a genetic decom-
position is a model of the development of those individual conceptions that align
with a concept.
3.2 Description of Mental Structures and Mechanisms
General descriptions of the mental structures and mental mechanisms that are used
in APOS Theory appear in this section. The construction of different mathematical
concepts in algebra, calculus, and statistics is used to illustrate how the structures
and the mental mechanisms that give rise to them develop.
Dubinsky (1991) discusses five types of reflective abstraction, or mental
mechanisms—interiorization, coordination, reversal, encapsulation, and generali-
zation—that lead to the construction of mental structures: Actions, Processes,
Objects, and Schemas. Figure 3.1 illustrates the relationships between these
structures and mechanisms (this figure also appears in Chap. 2). In Chap. 8 a
Fig. 3.1 Mental structures
and mechanisms for the
construction of mathematical
knowledge
18 3 Mental Structures and Mechanisms. . .
new version of this figure is presented, which takes into account the possible stage
of Totality.
The interaction of the elements in Fig. 3.1 can be described as follows:
. . . we consider that understanding a mathematical concept begins with manipulating
previously constructed mental or physical objects to form actions; actions are then
interiorized to form processes which are then encapsulated to form objects. Objects can
be de-encapsulated back to the processes from which they were formed. Finally, actions,
processes and objects can be organized in schemas. (Asiala et al. 1996, p. 9)
Dubinsky (1991) characterizes the overall relationship among these elements as
a “circular feedback system” (p. 106). Although the construction of mathematical
knowledge is nonlinear, as will be seen more clearly in Chap. 4, the APOS-based
description of the mental construction of a mathematical concept is presented in a
hierarchical manner. The depth and complexity of an individual’s understanding of
a concept depends on her or his ability to establish connections among the mental
structures that constitute it. These connections form the basis of a Schema
(described in more detail in Sect. 3.2.5 and in Chap. 7) whose coherence (described
briefly in Sect. 3.2.5 and in Chap. 7) is crucial to an individual’s ability to make
sense of mathematical situations related to the concept.
Assimilation and accommodation, the mechanisms used by Piaget in his work,
and the mechanism of generalization do not appear in Fig. 3.1. Assimilation of
knowledge refers to a mechanism by which a subject can apply a cognitive
structure, essentially without change, to include a cognitive object the subject has
not previously dealt with. Accommodation refers to a mechanism by which a
mental structure is reconstructed and modified in order to deal with a new situation.
Both mechanisms are related to the APOS idea of generalization.
In the rest of this section, each structure and the way it is constructed is described
and illustrated with examples.
3.2.1 Actions
According to Piaget and adopted by APOS Theory, a concept is first conceived as
an Action, that is, as an externally directed transformation of a previously
conceived Object, or Objects. An Action is external in the sense that each step of
the transformation needs to be performed explicitly and guided by external
instructions; additionally, each step prompts the next, that is, the steps of the Action
cannot yet be imagined and none can be skipped. For example, in the case of the
function concept, “an individual who requires an explicit expression in order to
think about the concept of function and can do little more than substitute for the
variable in the expression and manipulate it is considered to have an action
understanding of functions” (Dubinsky et al. 2005a, p. 338). Thus, the expression
acts as an external cue that indicates how the Action must be performed, step-by-
step, by the substitution of specific values.
3.2 Description of Mental Structures and Mechanisms 19
An individual who is limited to an Action conception relies on external cues.
In the case of the composition of two functions, an individual with an Action
conception would need to have explicit expressions for each function and could
only think about the composition for specific values. As Breidenbach et al. (1992)
note: an individual “would probably be unable to compose two functions in more
general situations, e.g., when functions had split domains, or if they were not given
by expressions at all” (p. 251).
Although the most primitive of structures (and often, the only one stressed in
traditional teaching), Actions are fundamental to APOS Theory. An Action con-
ception is necessary for the development of other structures. In particular, Processes
are interiorized Actions, and mental Objects arise because of the application of
Actions. New Actions lead to the development of higher order structures. For
instance, in the case of functions, performing operations on them spurs their
encapsulation as Objects.
Actions may be basic or complex depending on the context. Following are some
examples:
In Linear Algebra: The construction of the concept of n-tuple may begin by
performing the Action that consists in taking a specified quantity of numbers and
placing them in a particular order.
In Statistics: In the case of the mean, the Action of calculating the mean for a given
set of data is determined by the definition of the data set. Students who are learning
how to calculate the mean of a specific set of data make the calculations according
to the Actions prescribed by the formula, that is, by adding the values of pieces of
data and dividing the sum by the number of data pieces.
In Calculus: Actions are needed to construct an estimate of the definite integral as the
area under a curve: for example, in dividing an interval into specific subintervals of a
given size, constructing a rectangle under the curve for each subinterval, calculating
the area of each rectangle, and calculating the sum of the areas of the rectangles.
3.2.2 Interiorization and Processes
Processes are constructed using one of two mental mechanisms: interiorization or
coordination. Each of these mechanisms gives rise to new Processes. Interiorization
is explained here, and coordination is considered in Sect. 3.2.4.
As Actions are repeated and reflected on, the individual moves from relying on
external cues to having internal control over them. This is characterized by an
ability to imagine carrying out the steps without necessarily having to perform each
one explicitly and by being able to skip steps, as well as reverse them. Interioriza-
tion is the mechanism that makes this mental shift possible.
An action must be interiorized. As we have said, this means that some internal construction
is made relating to the action. An interiorized action is a process. Interiorization permits
one to be conscious of an action, to reflect on it and to combine it with other actions.
(Dubinsky 1991, p. 107)
20 3 Mental Structures and Mechanisms. . .
In the same spirit, Dubinsky et al. (2005a) give the following description of a
Process and interpret it for the case of functions:
As an individual repeats and reflects on an action, it may be interiorized into a mental
process. A process is a mental structure that performs the same operation as the action
being interiorized, but wholly in the mind of the individual, thus enabling her or him to
imagine performing the transformation without having to execute each step explicitly.
Thus, for example, an individual with a process understanding of function will construct a
mental process for a given function and think in terms of inputs, possibly unspecified, and
transformations of those inputs to produce outputs. (p. 339)
Although an Action and a Process, when related to a given concept, may involve
the same transformation, they differ in the following sense: for an Action, one must
actually make the transformation (either physically or mentally); for a Process
one can carry out the transformation without the need to go through each step.
Following are some examples:
In Linear Algebra: When the Actions involved in the construction of an n-tupleare interiorized into a Process, the subject can construct an n-tuple mentally even
when n is not specified; he or she can also consider the construction of n-tuples inany vector space, including infinite dimensional spaces. It is also possible for the
individual to think about the elements of the tuple, considering that the elements
may repeat, but the order in which they appear cannot be changed.
In Statistics: In the case of the mean, the Action of computing the mean of a set of
data points is interiorized into a Processwhen students can describe in general how to
compute the mean with being given a specific data set and grasp the idea that a mean
represents a characteristic of a set of numbers as a whole (Mathews and Clark 2003).
In Calculus: For the definite integral, the Action of determining the Riemann sum
for a particular partition is interiorized into a Process when an individual can
describe how the Riemann sum is determined for an unspecified partition and
imagine this process continuing with decreasing mesh size (maximum length of a
subinterval).
3.2.3 Encapsulation and Objects
Encapsulation occurs when an individual applies an Action to a Process, that is,
sees a dynamic structure (Process) as a static structure to which Actions can be
applied. Dubinsky et al. (2005a) offer the following explanation:
If one becomes aware of the process as a totality, realizes that transformations can act on
that totality and can actually construct such transformations (explicitly or in one’s imagi-
nation), then we say the individual has encapsulated the process into a cognitive object. Forthe function concept, encapsulation allows one to apply transformations of functions such
as forming a set of functions, defining arithmetic operations on such a set, equipping it with
a topology, etc. (p. 339)
3.2 Description of Mental Structures and Mechanisms 21
Examples of encapsulation of Processes include the following:
In Linear Algebra: Comparing n-tuples or performing binary operations on
n-tuples are Actions on n-tuples. For these Actions to be applied successfully, the
Process of forming an n-tuple must be encapsulated into an Object.
In Statistics: To think of the mean as one of several measures of central tendency
that gives information about a set of data and to ascertain its properties (which are
Actions), the Process of calculating a mean is encapsulated into a mental Object
(Mathews and Clark 2003).
In Calculus: The area under the curve for a function on a closed interval is the limit
of Riemann sums—an Action applied to the Riemann sum Process. In order to
determine the existence of this limit and/or to calculate its value, the student needs
to encapsulate the Riemann sum Process into an Object.
As reported in various APOS-based studies, the mechanism of encapsulation is
the most difficult. For example, in a study about students’ conceptions of the
fundamental theorem of statistics, Clark et al. (2007) reported:
However, moving beyond a process conception of mean is much more difficult. Three of
the students in this study had not progressed beyond a process conception of the mean.
Although they could perform the necessary actions, describe the process of computing the
mean of a set of numbers, and in some cases reverse this process, these students appeared
unable to conceive of the mean of a data set as an entity itself. They were unable to perform
any actions on the output of their processes or to associate any meaningful properties with
the means they computed. (p. 5)
In a study about the mental construction of two variable functions, Trigueros and
Martınez-Planell (2010) found that only one of their students had constructed an
Object conception.
Sfard also wrote about the “inherent difficulty of reification” (similar to encap-
sulation in APOS Theory), suggesting that:
The ability to see something familiar in a totally new way is never easy to achieve. The
difficulties arising when a process is converted into an object are, in a sense, like those
experienced during transition from one scientific paradigm to another. . . (Sfard 1991, p. 30)
As will be seen in Chap. 5, APOS-based instruction has had considerable success
in dealing with this difficulty.
3.2.4 De-encapsulation, Coordination, and Reversalof Processes
Once a Process has been encapsulated into a mental Object, it can be
de-encapsulated, when the need arises, back to its underlying Process. In other
words, by applying the mechanism of de-encapsulation, an individual can go back
to the Process that gave rise to the Object.
22 3 Mental Structures and Mechanisms. . .
The mechanism of coordination is indispensable in the construction of some
Objects. Two Objects can be de-encapsulated, their Processes coordinated, and the
coordinated Process encapsulated to form a new Object.
This is what happens mentally with function composition. To compose two
functions F and G to obtain F � G; the two function Objects must be
de-encapsulated to the Processes that gave rise to them. These Processes are then
coordinated, by applying the Process of F to the elements obtained by applying the
Process of G. The resulting Process is then encapsulated into a new Object.
As indicated in Fig. 3.1, a Process can be reversed. For example, Dubinsky
(1991) explained how the function Process can be reversed to obtain an inverse
function:
It is by reflecting on the totality of a function’s process that one makes sense of the notion of
a function being onto. Reflection on the function’s process and the reversal of that process
seem to be involved in the idea of a function being one-to-one. (p. 115)
The idea of a bijective function is constructed mentally and gives rise to an
inverse function by applying the mechanism of reversal.
In relation to function composition, the following three types of problems are
similar mathematically:
1. GivenF andG; findH such thatH ¼ F � G:2. GivenG andH; findF such thatH ¼ F � G:3. GivenF andH; findG such thatH ¼ F � G:
Cognitively speaking, however, they are different. According to Ayers et al.
(1988), in the solution of problems of the second and third type, “reversals of the
processes seem to be required” (p. 254). Dubinsky provides the following analysis
about the difference:
The first kind of problem [1] seems to require only the coordination of two processes that,
presumably, have been interiorized by the subject.
The second [2], howevermay require that the following be done for each x in the domain ofH.
2a. Determine what H does to x obtaining H(x).2b. Determine what G does to x obtaining G(x).2c. Construct a process that will always transform G(x) to H(x).
The third kind of problem [3] may be solved by doing the following for each x in the
domain of H.
3a. Determine what H does to x obtaining H(x).3b. Determine value(s) y having the property that the process of F will transform y to H(x).3c. Construct a process that will transform any x to such a y.
Comparing 2b with 3b (the only point of significant difference), we can see that 2b is a
direct application of the process of G whereas 3b requires a reversal of the process of F.It is perhaps interesting to note that this difference in difficulty (between [2] and [3]), which
is observed empirically and explained epistemologically, is completely absent from a
purely mathematical analysis of the two problems. They are, from a mathematical point
of view, the calculation ofH � G�1 and F�1 � G, respectively, which appear to be problems
of identical difficulty. This seems to be another important example in which the psycho-
logical and mathematical natures of a problem are not the same (cf. p. 113).
3.2 Description of Mental Structures and Mechanisms 23
Another situation in which relative difficulty can be explained by the requirement of
reversing a Process occurs in the development of children’s ability in arithmetic. According
to Riley, Greeno and Heller (1983, p. 157), “Problems represented by sentences where the
unknown is either the first ð?þ a ¼ b) or second ðaþ ? ¼ cÞ number are more difficult than
problems represented by equations where the result is the unknown ðaþ b ¼ ?Þ.” The firsttwo problem types involve a reversal of the Process, which, in the third type can be applied
directly. (Dubinsky 1991, p. 118)
Another example that Dubinsky (1991) presents about the generation of a new
process by the mechanism of reversal is related to integration:
A calculus student may have interiorized the action of taking the derivative of a function
and may be able to do this successfully with a large number of examples, using various
techniques that are often taught and occasionally learned in calculus courses. If the
process is interiorized, the student might be able to reverse it to solve problems in
which a function is given and it is desired to find a function whose derivative is the
original function. (p. 107)
The mechanism of coordination, in particular, how it is carried out mentally, is
currently under investigation. It is hypothesized that coordination of two Processes,
say PA and PB, can be thought of as the application of PA to PB (Fig. 3.2). For that to
be possible, the learner first needs to encapsulate PB into an Object, OB, in order to
be able to apply PA to it. Once that happens, the coordination can continue in the
following way: either OB is assimilated and PA can be applied to it, or PA is
accommodated so that the learner can apply it to OB. An alternative is for PB to be
applied to PA in a similar way. Whether coordination actually occurs in this way is
the subject of future study.
3.2.5 Thematization and Schemas
The interaction of the elements presented in Fig. 3.1 (Sect. 3.2) gives rise to
Schemas. According to Dubinsky (1991), a Schema is characterized by its dyna-
mism and its continuous reconstruction as determined by the mathematical activity
Fig. 3.2 Coordination of two Processes PA and PB
24 3 Mental Structures and Mechanisms. . .
of the subject in specific mathematical situations. The coherence of a Schema is
determined by the individual’s ability to ascertain whether it can be used to deal
with a particular mathematical situation. Once a Schema is constructed as a
coherent collection of structures (Actions, Processes, Objects, and other Schemas)
and connections established among those structures, it can be transformed into a
static structure (Object) and/or used as a dynamic structure that assimilates other
related Objects or Schemas.
For example, a Schema for vector space may include n-tuples and matrices as
Objects and polynomials and functions as Processes. All these structures may be
related by the fact that they share some properties, such as satisfying a set of
axioms that define a vector space. Coherence of this Schema lies in the mathe-
matical definition of vector space which the individual uses to determine whether
or not the Schema is applicable to a given situation. The construction of a
Schema as a mental Object is achieved through the mechanism of thematization.
This mechanism enables an individual to apply transformations to the Schema
structure.
Hence Schemas are structures that contain the descriptions, organization, and
exemplifications of the mental structures that an individual has constructed regard-
ing a mathematical concept. Studies that focus on the development of Schemas are
not very numerous and more research is needed to understand better how Schemas
develop and are applied. The development and application of Schemas as well as
their thematization are considered in greater detail in Chap. 7.
3.3 Overview of Structures and Mechanisms
Since mathematical concepts are not constructed directly, it is necessary for an
individual to construct mental structures to make sense of them (Piaget and Garcıa,
1983/1989). According to APOS Theory, individuals deal with mathematical prob-
lem situations by constructing and applying mental structures in their effort to
understand mathematical concepts. This involves transforming (via Actions or
Processes) previously established structures. These transformations then become
new Objects via the mechanism of encapsulation. APOS Theory is based on the
premise that an individual can learn any mathematical concept provided the
structures necessary to understand those concepts have been built (Dubinsky 1991).
Each of the structures that make up APOS Theory is constructed via a mental
mechanism: an Action is interiorized into a mental Process, a Process is
encapsulated into a cognitive Object, a Process can be reversed to construct another
Process, two Processes may be coordinated to form a new Process, and a Schema
can be thematized into a cognitive Object. In their study of uncountable infinite
Processes, Stenger et al. (2008) describe the terms structure and mechanism and the
relation between the two:
3.3 Overview of Structures and Mechanisms 25
Amental structure is any relatively stable (although capable of development) structure (i.e.,
something constructed in one’s mind) that an individual uses to make sense of mathemati-
cal situations. A source for a mental structure is a description of where that structure comes
from. A mental mechanism is a means by which that structure might develop in the mind(s)
of an individual or a group of individuals. (p. 98)
The constructions of mathematical knowledge described in this chapter illustrate
how making the most basic constructions is fundamental for an individual to
construct more robust structures. Mental structures and the mechanisms by which
they are constructed involve a spiral approach where new structures are built by
acting on existing structures. This idea is discussed in detail in Sects. 2.1 and 2.2.
Dubinsky (1997), referring to Piaget’s ideas, wrote that:
Objects, once constructed, can be transformed to make higher level actions and then
processes, and so on. This can continue indefinitely. Moreover, any action, process, or
object can be reconstructed, as a result of experiencing new problem situations on a higher
plane, interiorizing more sophisticated actions and encapsulating richer processes. The
lower level construction is not lost, but remains as a part of the enriched conception. (p. 98)
26 3 Mental Structures and Mechanisms. . .
Chapter 4
Genetic Decomposition
The ultimate goal of scientific research is the development of theories or models
to explain and/or to predict different kinds of phenomena (Woodward 2003).
As discussed in the previous chapters, research into students’ learning of mathe-
matics helps to predict what they may learn about a specific mathematical
concept and the conditions by which that learning takes place. This is an
important part of mathematics education as a research field and it is one of the
roles of APOS Theory.
Once the constructs of the theory are defined, models that show how those
constructs are related and developed. The models serve as the basis for working
hypotheses that can be tested experimentally. In APOS Theory the genetic decom-
position plays this role.
The role of the genetic decomposition as a hypothetical model of mental
constructions needed to learn a specific mathematical concept is the subject of
this chapter. Section 4.1 introduces what a genetic decomposition is, its importance,
and its predictive value. The complexities involved in the design of a genetic
decomposition are discussed in Sect. 4.2. In Sect. 4.3 the role of the genetic
decomposition in research is presented. Section 4.4 is a discussion of whether a
genetic decomposition is unique. The use of the genetic decomposition in the
design of teaching activities (further discussed in Chap. 5) and as a diagnostic
tool is considered in Sect. 4.5. Section 4.6 addresses the refinement of a genetic
decomposition. The chapter concludes with some examples of common errors and
misunderstandings about the genetic decomposition.
4.1 What Is a Genetic Decomposition?
A genetic decomposition is a hypothetical model that describes the mental
structures and mechanisms that a student might need to construct in order to learn
a specific mathematical concept. It typically starts as a hypothesis based on the
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_4,© Springer Science+Business Media New York 2014
27
researchers’ experiences in the learning and teaching of the concept, their
knowledge of APOS Theory, their mathematical knowledge, previously published
research on the concept, and the historical development of the concept. Until it is
tested experimentally, a genetic decomposition is a hypothesis and is referred to as
preliminary.A new mathematical concept frequently arises as a transformation of an
existing concept. As such a genetic decomposition consists of a description of
the Actions that a student needs to perform on existing mental Objects and
continues to include explanations of how these Actions are interiorized into
Processes. At this point, the concept is still seen as something one does. In
order to be conceived as an entity in its own right, something that can be
transformed, the Process is encapsulated into a mental Object. It is entirely
possible that a concept may consist of several different Actions, Processes, and
Objects. A genetic decomposition may include a description of how these
structures are related and organized into a larger mental structure called a
Schema. Included in the description of a Schema may be an explanation of
how the Schema is thematized into an Object. The genetic decomposition also
explains whatever is known about students’ expected performances that indicate
differences in the development of students’ constructions.1
In addition to describing how a concept might be constructed mentally, a genetic
decomposition might include a description of prerequisite structures an individual
needs to have constructed previously, and it might explain differences in students’
development that may account for variations in mathematical performance. Thus, a
genetic decomposition is a model of the epistemology and cognition of a mathe-
matical concept (Roa-Fuentes and Oktac 2010).
A preliminary genetic decomposition can guide the development of an instruc-
tional treatment (described in Chap. 5). Implementation of the instruction
provides an opportunity for gathering data, usually in the form of written
instruments and/or in-depth interviews. In analysis of the data, two questions
are asked: (1) Did the students make the mental constructions called for by the
genetic decomposition? (2) How well did the subjects learn the mathematical
content? Answers to these questions may lead to revision of the genetic decom-
position and/or the instruction. At this point, the genetic decomposition is no
longer regarded as preliminary. Further refinements are possible, as each
1 Some researchers use the term “cognitive path” to describe a specific ordering of concepts that
students seem to follow when learning a mathematical topic. Cognitive paths are found by means
of a specific statistical method using data from students (Vinner and Hershkowitz 1980).
A cognitive path describes a process of learning focused on the mathematical aspects of the
concept. Although it may seem that there is some similarity between “cognitive path” and “genetic
decomposition,” their focus and content are different.
Cognitive paths describe or suggest a linear cognitive progression based on an analysis of the
mathematical aspects of the concepts involved. Instead of a linear progression, a genetic decom-
position in APOS Theory describes the mental structures and the mechanisms by which those
structures are constructed. Confusion with cognitive paths may explain some of the errors
discussed later in this chapter.
28 4 Genetic Decomposition
refinement leads to further revision of the instruction, which provides an
opportunity for new data analysis. Ideally, the cycle of refinement ! revision
! data analysis yields a genetic decomposition that reflects very closely the
cognition of the concept for many individuals and that can be used in the design
of instruction that positively affects student learning.
There are several issues to be clarified. When referring to a student in the context
of APOS Theory, the researcher is not considering a specific student. Rather, he or
she is considering a “generic student,” that is, a representative of the class of
students who are learning that concept. Also, it may seem that a genetic decompo-
sition is unique. However, as pointed out in Sect. 4.4, the genetic decomposition for
a given mathematical concept may not be unique. What is important is that a
genetic decomposition predicts the mental constructions deduced from the analysis
of data gathered in experimental designs (Dubinsky 1991).
Because the genetic decomposition of a concept is described linearly, it might
seem as though the concept develops linearly. However, this is mostly a conse-
quence of the description, which does not reflect the possibility of different
trajectories that include starts, stops, and discontinuities that occur in learning.
In addition, APOS Theory does not rule out the possibility that mental structures,
once developed, may not always be applied when called for. Thus, a genetic
decomposition does not explain what happens in an individual’s mind, as this is
probably unknowable; predict whether an individual will apply a given structure
when called upon; nor offer an exclusive theoretical analysis of how mathematics
is learned. APOS theory acknowledges that a student may pursue different
learning paths or follow different trajectories, as a student moves from Process
to Action and back to Process or from Object to Process and back to Object.
Despite individual differences, a genetic decomposition describes the structures a
student needs to construct in her or his learning of a concept. When verified
empirically, a genetic decomposition can serve as a useful model of cognition, as
evidenced by a number of empirical studies that show the efficacy of APOS
theory as a tool for describing students’ conceptions and in the design of effective
instruction (Weller et al. 2003).
Examples of genetic decompositions appear throughout the remainder of this
chapter. Examples in this section show preliminary genetic decompositions. Later
in the chapter refinements of some genetic decompositions, based on research data,
will be shown.
4.1.1 A Genetic Decomposition for Function (Based on Ideasfrom Dubinsky 1991)
The construction of the function concept starts with Actions on a set. Given a set of
numbers or other kinds of elements, these Actions involve taking an element from
one set, explicitly applying a rule, typically an algebraic formula, and assigning to
4.1 What Is a Genetic Decomposition? 29
that specific element a unique element from the second set. As these Actions are
performed on different sets, say ordered pairs, points, or non-numeric objects, the
individual reflects on them and perceives them as a dynamic transformation. At this
point, interiorization starts, as the individual begins to see a function as a type of
transformation that pairs elements of one set, called the domain, with elements of a
second set, called the range. This means that the individual has constructed a mental
structure that performs the same transformation as the Action, but wholly in the
mind of the individual.
An individual who shows a Process conception of function can think of a
function in terms of accepting inputs, manipulating them in some way, and produc-
ing outputs without the need to make explicit calculations. Evidence of a Process
conception of function might include the ability to determine whether a function
has an inverse, which would require a reversal of the function Process, or to
describe how one would compose two functions, which would require a coordina-
tion of two function Processes.
Applications of Actions or other Processes applied to the function Process
lead to its encapsulation as a cognitive Object. The mechanism of encapsulation
moves the learner’s focus away from the concept of function as a dynamic trans-
formation to a static entity that itself can be examined and transformed. Indications
of encapsulation might include an individual’s ability to form sets of functions,
or to perform arithmetic operations on functions, or to construct a function that
is a limit of a sequence of functions. In the first, a function is treated as an
element; in the second, as an input to binary operations; and in the third, as the
transcendent Object of an infinite Process that produces a sequence of functions.
In each of these cases, functions are treated as static entities to which Actions can
be applied.
An individual who can determine whether the relationship between two entities
defines a functional relationship, and can coordinate various Processes to determine
the domain and range of a function, may be giving evidence of constructing a
function Schema. An indication of the coherence of a function Schema would
include an individual’s ability to determine whether a particular mathematical
situation defines a functional relationship.
4.1.2 A Genetic Decomposition for Induction (Dubinsky1991, pp. 109–111)
The genetic decomposition for the induction Process assumes prior construction
of the function and logic Schemas. The function Schema includes a Process for
evaluating a function for a given value in its domain. The logic Schema includes
the ability to construct statements in the first-order propositional calculus. For
instance, the individual constructs a Process for logical necessity; that is, in
certain situations, he or she will understand that if A is true, then, of necessity,
Bwill be true.The mental development of induction from this initial starting point is
illustrated in Fig. 4.1 and is described in detail in the remainder of this subsection.
30 4 Genetic Decomposition
In general, the first-order propositions in the logic Schema described above are
Processes that originate from interiorizing Actions (conjunctions, disjunctions,
implications, negations) on declarative statements (Objects). For example, the
formation of the disjunctionP _ Q can be described as an Action on the statements
P; Q. The Action, which goes well beyond simply putting these symbols into a
disjunction expression, involves the construction of a mental image of the two
statements and the determination of the truth or falsity of the disjunction in
various situations. Through interiorization, the Action is transformed into a Process
for forming the disjunction of two declarative statements. If nothing further is done
after this Action is interiorized, then it will be impossible for the disjunction to be
combined with other statements. Specifically, to combine the disjunction P _ Q with
a declarative statement R to form the statement ðP _ QÞ ^ R, the disjunction Processmust be encapsulated to form a new Object ðP _ QÞ to which the statement R can be
conjoined. Thus, in general, the Actions of conjunction, disjunction, implication, and
negation that are applied to declarative statements must be interiorized and then
encapsulated in order to construct more complex first-order propositions as Objects
[note how the use of parentheses in mathematical notation corresponds here to
encapsulation (Dubinsky and Lewin 1986)]. Iterating this procedure, the subject
enriches her or his logic Schema to obtain a host of new Objects consisting of first-
order propositions of arbitrary complexity.
Fig. 4.1 Genetic decomposition for the concept of mathematical induction (Dubinsky and
Lewin 1986)
4.1 What Is a Genetic Decomposition? 31
The function Schema must be accommodated to enable the construction of a
Process that transforms positive integers into propositions, that is, to obtain
a proposition-valued function of the positive integers. Consider, for example, a
statement such as “given a number of dollars it is possible to represent it with
$3 chips and $5 chips.” To understand the meaning of such a statement, the subject
must construct a Process in which the “number of dollars” (in the original statement)
is replaced by that value of n. This is a proposition-valued function. In order to
evaluate it, the subject must construct another Process, whereby given n a search
is made to determine whether it is possible to find nonnegative integers j, k such thatn ¼ 3jþ 5k . It is useful for the subject to discover that the value of this function
is true for n ¼ 3; 5, false for n ¼ 2; 4; 6; 7, and true for all higher values.
Steps in the construction of proof by induction include the encapsulation of the
Process of implication, which then becomes an Object that is in both the domain
and the range of a function. This is followed by further assimilation of the function
Schema to include implication-valued functions, as well as the interiorization of a
Process of going from a proposition-valued function of the positive integers to its
corresponding implication-valued function.
The logic Schema needs to include a Process called modus ponens. This
Process is the interiorization of an Action applied to implications (assuming, as
above, that they have been encapsulated into Objects). The Action begins with
the hypothesis, determining that it is true, followed by asserting the truth of the
conclusion. These constructions make it possible for a student to coordinate the
function Schema, as it applies to an implication-valued function Q (obtained
from a proposition-valued function P), and the logic Schema, as it applies to the
Process modus ponens. This leads to construction of a function n ! QðnÞ, wheren is a positive integer and Q is the implication-valued function that transforms nto PðnÞ ) Pðnþ 1Þ. For an integer n in the domain of positive integers, one
determines the value of the function Q, which, in this case, involves determina-
tion of the truth or falsity of PðnÞ ) Pðnþ 1Þ. If it has been established that Qhas a constant value of true, the first step in this new Process is to evaluate P at
1 and to determine that P(1) is true (or more generally to find a value no such that
PðnoÞ is true). Next the function Q is evaluated at 1 to obtain Pð1Þ ) Pð2Þ .Applying modus ponens and the fact (just established) that P(1) is true yields theassertion P(2). The evaluation process is again applied to Q, but this time with
n ¼ 2, to obtain Pð2Þ ) Pð3Þ. Modus ponens again gives the assertion P(3). Thecycle is repeated ad infinitum, alternating the Processes of modus ponens and
evaluation. This produces a rather complex coordination of two Processes that
leads to an infinite Process. This infinite Process is encapsulated and added to the
proof Schema as a new Object, proof by induction.
In the presentation of this genetic decomposition, Dubinsky noted that the
student will not necessarily be aware of these Schemas, that is, the subject will be
able to think in terms of plugging a value of a positive integer into a statement to ask
if the result is a true statement, but will not necessarily be aware of the fact that he
or she is working with a proposition-valued function or an implication-valued
function. The development of the student’s function Schema can only be inferred
32 4 Genetic Decomposition
from her or his Actions (Dubinsky 1991). In a similar vein, Dubinsky points out that
“[i]n describing this construction we reiterate the point that, in the context of this
theory, it is never clear (nor can it be) whether we are talking about a schema that is
present or a schema that is being (re-)constructed” (p. 112).
It is interesting to note that this genetic decomposition reveals a cognitive step,
which research has pointed out as providing a serious difficulty for students, that is
not apparent when considering induction from a purely mathematical point of view.
Specifically, if P is a proposition-valued function to be proved, P(n) can be any
proposition, in particular, it can be an implication. From a mathematical point of
view, there is nothing new in the proposition-valued function Q defined by
QðnÞ ¼ PðnÞ ) Pðnþ 1Þ;
that is, once one understands P, then, as a special case, one understands Q.However, this is not the case from a cognitive point of view: implications are the
most difficult propositions for students and generally the last to be encapsulated.
Furthermore, there is a difference between constructing P from a given statement
and constructing Q from P. From a cognitive perspective, the construction of Qfrom P “is a step that must be taken” (Dubinsky 1991, p. 109).
4.2 The Design of a Genetic Decomposition
One question that arises when considering genetic decompositions is their design.
Specifically, how are they designed and what is needed in their design? A prelimi-
nary genetic decomposition can arise in one of several ways. The preliminary
genetic decomposition for a particular concept is based on the researchers’ mathe-
matical understanding of the concept, their experiences as teachers, prior research
on students’ thinking about the concept, historical perspectives on the development
of the concept, and/or an analysis of text or instructional materials related to the
concept. Following are some examples.
Some preliminary genetic decompositions are designed by taking into account
mathematical descriptions of a concept, together with the researchers’ experiences
as learners or teachers. This is the case for the genetic decomposition for induction
described in Sect. 4.1.2. Others are designed from data from previous mathematics
education research, not necessarily conducted using APOS theory, into students’
difficulties in learning a particular concept. This is the case for spanning set and
span that is discussed in Sect. 4.2.1. It may be recalled here that the description
included in the genetic decomposition is not the same as the mathematical intro-
duction of a concept, as is clearly illustrated by the example on mathematical
induction shown in Sect. 4.1.
There are genetic decompositions based on data from observations of students
who are learning a mathematical concept. Analysis of observations leads to a
description of the cognition of the concept, which can be verified empirically.
4.2 The Design of a Genetic Decomposition 33
Trigueros and Martınez-Planell (2010) followed this approach in their study of
students’ learning of two-variable functions. The authors used observations of
student work to develop a preliminary genetic decomposition of two-variable
functions. They then conducted interviews with students who had completed the
course and used the interview data to refine their preliminary analysis.
The design of a genetic decomposition can also be based on the historical develop-
ment of the concept. A study of the historical development of a concept may point to
mental constructions that individuals might make. In response to historical arguments
against the existence of actual infinity, Dubinsky et al. (2005a, b) used APOS Theory
to explain how potential and actual infinity represent two different conceptualizations
linked by the mental mechanism of encapsulation. Although their explanation did not
lead directly to a genetic decomposition on mathematical infinity, it strongly
influenced the design of a genetic decomposition on infinite repeating decimals
(Weller et al. 2009, 2011; Dubinsky et al. 2013) and informed a study of infinite
iterative Processes conducted by Brown et al. (2010). In the former case, the prelimi-
nary genetic decomposition was hypothetical; it was later tested empirically by
analyzing data from students who completed specially designed instruction based on
the genetic decomposition. In the latter case, the preliminary genetic decomposition
was empirical; it arose from an analysis of interview data involving students who tried
to solve a problem in set theory. In their efforts tomake sense of the problem situation,
the students constructed a variety of iterative Processes. Brown et al. (2010) used their
understanding of the historical development of the concept of mathematical infinity to
guide their analysis of the interviews that culminated with a genetic decomposition of
infinite iterative processes.
Text materials may also inform the design of a preliminary genetic decomposi-
tion. Specifically, for a given concept the didactical approach used in the text can
help researchers to determine how students might come to understand the concept.
This analysis is complemented by researchers’ descriptions of the mental structures
students need to construct in order to succeed in learning those concepts. This will
be exemplified in the discussion of linear transformations in Sect. 4.4.
Finally, genetic decompositions can be developed from data. In this case,
students are interviewed and the transcription of the interview is divided into
small pieces. By comparing these pieces, it is possible to find differences in
students’ performance on specific tasks. Differences in performance may uncover
instances where certain mental constructions need to be made. Lack of success in
completing a task may indicate that the student has not made the needed mental
constructions while success with the task may uncover evidence that those mental
constructions have been made. The totality of the results obtained by this type of
analysis leads to the organization of the mental constructions that make up the
genetic decomposition.
As a result of one or more of these methods, the genetic decomposition can be a
simple model of the main constructions the researchers think are needed in order to
learn a concept, as illustrated in Sect. 4.2.1. It can also be a model where many of
the complexities involved in the construction of the concept are described. This is
exemplified in Sect. 4.2.3.
34 4 Genetic Decomposition
In Sect. 4.1, two examples of genetic decompositions were presented; the goal
was to illustrate what a genetic decomposition is and how APOS constructions are
involved in describing the mental constructions involved in learning a concept.
In Sect. 4.2.1 an additional example, involving the concepts of spanning set and
span, is introduced in order to show how previous research informs the develop-
ment of a preliminary genetic decomposition.
4.2.1 Genetic Decomposition for Spanning Set and Span(Based on Ku et al. 2011)
The design of this genetic decomposition was informed by three sources of data:
results from the review of literature about the learning of basis and spanning set, a
report on students’ difficulties in construction of the concept of basis (Ku et al.
2011), and evidence (also from Ku et al. 2008) of differences in students’
constructions of the concepts of spanning set and span.
Results of the literature review for Ku et al. (2008) indicated the importance of
basis as a concept related to vector space and students’ tendency to have a
conceptual image of spanning set as a basis. Ku et al. (2008) found that students
were not able to differentiate between the concepts of basis and spanning set, had
difficulty working with vector spaces different from R2 and R3, and struggled with
the concept of span. These findings suggested the need for certain prerequisite
constructions. Specifically, in order to construct the concepts of spanning set and
span, students need to recognize that different types of sets, such as n-tuples,polynomials, and matrices, are vector spaces and that a vector space can be
generated by spanning sets of different sizes.
The analysis in Ku et al. (2008) also revealed that students who constructed an
Action conception of the concept of basis were able to perform Actions to construct
linear combinations using given vectors but were unable to interpret span as the set
of all the linear combinations of the basis vectors. This fact was taken into account
in designing the genetic decomposition for spanning set and span. Specifically, one
starts with Actions involving the formation of linear combinations. These Actions
are interiorized into a Process so that the individual can imagine all linear
combinations of a given set of vectors. A reversal of this Process enables the
individual to determine whether an arbitrary vector in a vector space can be written
as a linear combination of the vectors in the given set.
Ku et al. (2008) found that many of the students in their study had difficulties
with the concept of set, especially when performing Actions on sets of vectors
whose elements included parameters. When solving systems of equations, these
students could perform Actions involved in manipulation of the equations in order
to find the solution of the system but could not differentiate between parameters
as general numbers and unknowns of the equations. Students with a Process
conception of basis were able to relate this concept to a vector space, showed
fewer problems in interpreting solution sets of systems of equations, and
4.2 The Design of a Genetic Decomposition 35
interiorized the Process for forming a basis. However, with respect to the latter,
they encountered difficulties. In particular, they could not differentiate the concept
of basis from that of spanning set and had trouble interpreting the span of a given
basis. The reason for this difficulty lies in their inability to make sense of the
solution of a system of equations Ax ¼ b when b has general numbers or
expressions as elements. These difficulties were taken into account in designing
the genetic decomposition. Specifically, in the description of prerequisite concepts,
it is stated explicitly that students need to have in their minds the concepts of
solution set and variable as Objects.
4.2.2 Prerequisite Constructions
The mental constructions of spanning set and span assume construction of the
concepts of vector space, variable, and solution set of a system of linear equations.
With respect to vector spaces, students need to demonstrate the ability to work with
familiar examples, such as spaces of n-tuples with real-valued entries, and recog-
nize that other types of sets, such as sets of polynomials and sets of matrices, are
vector spaces.
The solution set of a system of equations plays an important role in development
of the concepts of spanning set and span. Specifically, to determine whether a
subspace of a vector space is spanned by a given subset of a vector space, or to
verify that a particular vector lies in the space generated by that subset, one needs to
solve a system of equations. Therefore, students need to have constructed this
concept as an Object.
The concept of variable plays an important role in the interpretation of solution
sets as spanning sets and spans. Specifically, a subspace generated by a subset of a
vector space may be represented analytically in terms of a generalized vector
involving variable expressions. Therefore, students need to work with variables
as mathematical Objects in order to understand variables as unknowns, general
numbers, parameters, or variables in functional relationships and to move flexibly
among these interpretations (Trigueros and Ursini 2003).
4.2.3 Mental Constructions
Given a vector space V with a specific scalar field K, students perform Actions on a
given subset S of vectors from V, specifically the Action to construct linear
combinations with vectors from S and scalars from K. These Actions consist of
multiplying vectors by scalars and summing the result of the multiplications to
obtain a new vector in V. Interiorization of these Actions yields a Process for
constructing a new vector which is an element of the vector space, that is, the
36 4 Genetic Decomposition
Process of constructing a linear combination. The reversal of this Process allows the
student to verify if a given vector can be written as a linear combination of a given
set of vectors. Students who show they have constructed these processes are
considered to have a Process conception of a linear combination.
By coordinating the reversal of the Process of constructing the set of all linear
combinations of vectors in a subset S of V with the Process for finding the solution
set of a system of equations, the learner can verify the existence of scalars in K that
can be used to determine whether vectors in a subset T of V can be expressed as
linear combinations of S. In short, this coordination enables the learner to determine
whether a subset T of V is generated by the subset S.When different sets are compared and considered as different possible spanning
sets for a set of vectors T, the coordinated Process mentioned above is encapsulated
into an Object called spanning set. A student who has an Object conception of
spanning set can make comparisons to decide whether a given vector space can be
generated by different spanning sets independently of the size of the set or their
specific elements.
When the Process for construction of a set T generated by S is coordinated with
the Process for vector spaces, the learner can verify that T is a vector space. This last
Process is encapsulated into an Object that can be called generated space, spanned
space, or span, of the original set of vectors S. These constructions enable studentsto differentiate between the concepts of span and spanning set.
This analysis does not ignore those concepts that are related to spanning sets,
such as linear independence or dependence, basis, and dimension. It includes
consideration of how the construction of a spanning set can help students under-
stand concepts like those that are related to it, or if there are difficulties in the
construction of this concept that act as obstacles when relating it with other linear
algebra concepts (Ku et al. 2011).
The genetic decomposition does not end with the construction of Objects. It
should also include a description of how links among different Actions, Processes,
and Objects can develop as a Schema. The mental development of a Schema will be
discussed in Chap. 7.
As research has evolved, there are many genetic decompositions published by
authors around the world (see the annotated bibliography in Chap. 12). In general
these genetic decompositions have been tested by research and either supported or
refined and then used for new research or in the design of teaching activities.
Refinements of genetic decompositions will be considered in greater detail in
Sect. 4.5.
4.3 Role of the Genetic Decomposition in Research
Genetic decompositions play a central role in APOS-based research, since a
theoretical model is necessary to provide researchers with hypotheses that can
serve as the basis for the design of theory-based instruments to obtain and analyze
4.3 Role of the Genetic Decomposition in Research 37
data from students. Students’ constructions can be deduced from their work and
their responses to interview questions compared with what is predicted in the
preliminary genetic decomposition. A genetic decomposition acts as a lens, analo-
gous to a diffraction grating2 that researchers use to explain how students develop,
or fail to develop, their understanding of mathematical concepts. For example, on a
given task, one student may perform the task correctly, another may have difficulty,
and still another may completely fail. The genetic decomposition may explain
discrepancies in performance. The student who succeeds may give evidence of
having successfully made one or more of the mental construction(s) called for by
the genetic decomposition. The student who shows limited progress may show
evidence of having begun to make the construction(s). The student who fails may
not have made the construction(s) at all or may give evidence of having been
unsuccessful in having made the necessary construction(s). If the differences in
student performance cannot be explained by the genetic decomposition, then it may
be the case that the genetic decomposition needs revision. Thus, on one hand, the
genetic decomposition guides the analysis; on the other, it points out gaps in the
researchers’ understanding of how the concept develops in the mind of the individ-
ual. Either way, a genetic decomposition is a tool by which researchers try to make
sense of how students go about learning a concept and to explain the reasons behind
student difficulties. Moreover, use of a theoretical model increases the reliability of
the analysis, provides a means to describe student thinking, and serves as a
diagnostic and predictive tool.
When using a genetic decomposition, different researchers can analyze the same
data and obtain comparable results. Working as a team they can interpret their
results in terms of the model. Without a model they might have a difficulty
agreeing on or negotiating their interpretations. Thus, the analysis of data becomes
more reliable when it is based on a theoretical model such as a genetic
decomposition.
As described in Asiala et al. (1996), a genetic decomposition needs to be tested
experimentally. The goal is to test the validity of the model: Did the students make
the mental constructions called for by the theoretical analysis? If the constructions
described in the genetic decomposition are observed, the model is supported. If the
students appear to construct the concept in a way that differs from what is described
in the genetic decomposition, then the model is refined or, if the discrepancies are
too great, discarded in favor of a new genetic decomposition.
The following example illustrates how the analysis of data may lead to a
refinement. In an interview in a study of students’ understanding of spanning sets
and span (Ku et al. 2011), the authors observed that many students did not understand
that the vectors that make up a spanning set are elements of the subspace generated
by that set. This difficulty was attributed to a mental construction not accounted
2A diffraction grating is an instrument used to analyze the light coming from stars. It decomposes
the incoming light by diffraction to obtain a pattern of colored lines. These patterns allow
researchers to know the chemical composition of the star.
38 4 Genetic Decomposition
for in the preliminary genetic decomposition. This difficulty arose in students’
responses to the following interview item taken from a linear algebra textbook:
Let v1 ¼1
0
0
24
35; v2 ¼
0
1
0
24
35and let H ¼
ss0
24
35 s 2 R
8<:
9=;:
Therefore each vector of H is a linear combination of fv1; v2g since
ss0
24
35 ¼ s
1
0
0
24
35þ s
0
1
0
24
35
Is fv1; v2g a spanning set for H?
Carlos3 was among the students who had difficulty with this item. The following
excerpt exemplifies this:
Carlos: Yes. All possible linear combinations of v1 and v2 span H. Neither v1 nor v2 can generatethe third element, but H doesn’t have it either, so it is not necessary. . . [then he explains]It’s a spanning set for H because if we take all the possible linear combinations in the
reals, then clearly we can see that it can be any number. . . well, any number in H and for
example H doesn’t have. . . it has a zero in the third element so, no, well. . .it would be,
it’s not needed and we see it here. . . I mean none of the two has it so. . . If H had another
s here for example [he refers to the vector ðs; s; 0Þ], it wouldn’t be a spanning set for H,we would need another vector, which had for example, I don’t know. If it were linearly
independent and if it had an element in the last position, but since these two don’t have it,
but H doesn’t either, then H can be spanned by these two vectors.
Carlos may have grasped the idea of the span of a set being formed by
“all possible linear combinations.” However, it is not clear if he considers whether
or not the vectors v1 and v2 belong to H. The interview continues with the following
question, in order to provoke more reflection (Fig. 4.2):
I: Can you give another spanning set for H?
Carlos: (writes)
Fig. 4.2 Carlos finds another spanning set
3 All the names of interview subjects are pseudonyms. The interviewer’s words are identified with
“I:” throughout the text. “I” does not denote a single individual, as there were different
interviewers for different studies, and sometimes multiple interviewers for the same study or
even the same interview.
4.3 Role of the Genetic Decomposition in Research 39
I: Let’s see. Why is this a spanning set?
Carlos: Because they are two linearly independent vectors and if we take any numb. . . I mean this
is in the reals, so if we take any number inH, well for example inH, I don’t know, for it tobe 1 and 1.Wemultiply this one by 1/5 added to this one multiplied by 1/3 and it spansH.
Then the interviewer asks Carlos to explain how he would find a spanning set
for H if the question didn’t provide the set {v1, v2}. Carlos responds as follows:
Carlos: If I didn’t have this? Well, s is in the reals, so it could be any number. Well, it would
be enough to take two vectors that, I mean, with which I can generate a real number
in the first one and a real number in the second one and that would be enough.
Like many of the students, Carlos did not yet understand that the elements of a
spanning set necessarily belong to the span. The researchers observed that this
student, and many others, had not made a construction related to the necessity of the
vectors belonging to H. The researchers concluded that one of the reasons for the
difficulty lay in the students’ failure to perform the Action of finding the span of
different spanning sets and of determining whether the elements of the spanning set
are included in the span. This observation helped the researchers to guide the
students to make the necessary construction, either during the interview or later
in class. This finding led the researchers to make a refinement to the preliminary
genetic decomposition.
The possibility of predicting the constructions students need in their learning
of mathematical concepts provides researchers with a useful tool in the design of
activities and teaching sequences that help students to make the constructions called
for by the theoretical analysis. The way in which a genetic decomposition guides
the design of instruction is considered in Chap. 5.
4.4 A Genetic Decomposition Is Not Unique
It has been reiterated several times in this book that a genetic decomposition is
not unique, that is, it does not provide a single way in which all students construct
a specific mathematical concept. Instead, it serves as a theoretical model which
may help in understanding those constructions that appear in most students’ work.
Although it describes a possible trajectory for the construction of the concept,
APOS Theory acknowledges that different students can follow paths different
from those described in a particular genetic decomposition. Thus, the value of
a genetic decomposition resides in its use as a general model which describes
those constructions that are found to be needed by most of the students in the
learning of a concept.
As with any general and descriptive theoretical model, several genetic
decompositions can be designed by different researchers or even by the same
group of researchers to describe the learning of a particular concept. If those
genetic decompositions are supported by empirical studies of students’
40 4 Genetic Decomposition
constructions, they could all be considered reasonable descriptions of students’
constructions. So far, there are only a few examples of different preliminary
genetic decompositions for the same concept, and in the case of genetic
decompositions supported by the results of APOS-based instruction, no diversity
has been found. Of course one would expect to find some agreements among
different supported models, but they may also have differences. Issues involving
differences in a genetic decomposition, and how those differences may be
resolved, is a subject for further research.
One exception is the work of Roa-Fuentes and Oktac (2010), who proposed
two different preliminary genetic decompositions of the linear transformation
concept. The first of these aligns with how this concept is typically taught and
how it appears in textbooks. The second one takes into account a suggestion
reported in the literature (Dreyfus et al. 1999) that is based on instruction where
nonlinear transformations are considered before the introduction of linear
transformations. In Roa-Fuentes and Oktac (2010), only Action, Process, and
Object constructions were investigated; Schema did not form part of the study.
As the word preliminary indicates, these theoretical analyses emerged before the
collection of any data.
In Sects. 4.4.1–4.4.4, these two genetic decompositions are discussed and
explained in detail. After considering the prerequisites, a general description is
accompanied by the presentation of a figure that summarizes the structures that
need to be constructed and the relations among those structures. Since the two
theoretical analyses differ only in how the Processes of the properties of linearity
are constructed, the two genetic decompositions are considered separately up to this
point. After this point, the two genetic decompositions are identical.
4.4.1 Prerequisites for the Construction of the LinearTransformation Concept
Since linear transformations are defined as functions between vector spaces, the
linear transformation concept depends on prior construction of a function Schema
and vector space Object. In order to define a function whose domain and range are
vector spaces, the function Schema needs to assimilate the vector space Object.
Since the test of whether a transformation is linear involves working with linear
combinations, it is necessary for the learner to have constructed the concept of
linear combination as a mental Object.
4.4.2 Genetic Decomposition 1
In the construction of the linear transformation concept according to this genetic
decomposition, an individual starts by applying Actions to specific vectors from a
4.4 A Genetic Decomposition Is Not Unique 41
particular vector space. Specifically, the individual checks the addition property of
linearity by applying the rule of assignment given by a particular transformation to
compare the image of the sum of two vectors with the sum of their images. Limited
to an Action conception, an individual cannot think beyond specific vectors or
specific transformations.
As the individual reflects on these Actions and begins to think in general about
this property for all the vectors in the domain space, without the need to work with
specific vectors and without the need to make specific calculations, these Actions
are considered to have been interiorized into a Process—the addition property of
linearity of a transformation between two vector spaces.
In the preliminary genetic decomposition, as shown in Fig. 4.3, Roa-Fuentes and
Oktac (2010) considered the possibility of an intermediate step between an Action
conception and a Process conception. Roa-Fuentes and Oktac (2012) found that this
intermediate step—checking the addition property of linearity for an arbitrary pair
of vectors without consideration of all of the vectors in the domain—was not
supported by empirical analysis.
As illustrated in Fig. 4.3, the Process of the scalar multiplication property of the
linearity of a transformation between two vector spaces is constructed similarly.
4.4.3 Genetic Decomposition 2
This genetic decomposition begins with construction of the concept of a
(general) transformation between two vector spaces. This concept has to have
been constructed as an Object, since determination of the preservation of vector
Fig. 4.3 Construction of the properties of linearity of a transformation between two vector spaces
as Processes by interiorizing Actions on vectors (Roa-Fuentes and Oktac 2010, p. 105)
42 4 Genetic Decomposition
addition and of scalar multiplication under the transformation are Actions applied
to the transformation.
The transformation Object is de-encapsulated so that the underlying Process can
be utilized. This Process is coordinated with the Process of the binary operation
“vector addition” through the universal quantifier 8 to generate a new Process for
determining whether the transformation satisfies the property of addition for all
pairs of vectors in the domain. The Process related to the transformation allows the
individual to think of the images of the domain vectors under the transformation.
The Process related to vector addition allows the individual to form sums of vectors
in the domain and in the range. By coordinating these two Processes, the individual
can do two things: (1) form a sum of any two vectors in the domain and apply the
transformation to that sum and (2) find the images of any two vectors in the domain
and add them. Determination of the sum property involves comparison of the
results obtained from (1) and (2) for all the vectors in the domain.
A similar coordination occurs for scalar multiplication. This is illustrated in
Fig. 4.4.
The principal difference between the two genetic decompositions lies in the
treatment of the transformation. In Genetic Decomposition 1, the transformation is
applied to a pair of vectors (and their sum). Through interiorization, this Action
is extended to all pairs of vectors in the vector space. In Genetic Decomposition 2,
the binary operations of vector addition and scalar multiplication are Processes
applied to the transformation Object, which is subsequently de-encapsulated so that
it can be coordinated with those Processes.
Fig. 4.4 Construction of the properties of linearity as Processes starting with transformation as an
Object
4.4 A Genetic Decomposition Is Not Unique 43
4.4.4 Genetic Decompositions 1 and 2: Constructing Processand Object Conceptions of Linear Transformation
Whichever genetic decomposition is followed in the construction of the two
Processes in Sects. 4.4.2 and 4.4.3, once constructed, these two Processes of
verifying the two linearity properties are coordinated via the logical connector
“and” to give rise to a new Process. The new Process is constructed when the
individual can think of the two Processes simultaneously, as in preserving linear
combinations of vectors under a linear transformation, as shown in Fig. 4.5.
Encapsulation occurs with the need to apply Actions. With a conception of linear
transformation as a mental Object, an individual can perform operations, such as
adding or composing two linear transformations, and can ask questions about the
properties linear transformations may or may not satisfy. For example: Under what
conditions is a linear transformation invertible?
4.5 Refinement of a Genetic Decomposition
When researchers design a genetic decomposition, it must be tested empirically.
The analysis may lead to mixed results. Some of the constructions predicted by the
preliminary analysis may appear to have been made, or seem reasonable given the
data, others may be lacking or different from those proposed, and others, not
accounted for by the preliminary analysis, may surface as a result of the analysis.
When this happens, the genetic decomposition needs to be refined to reflect what
has been found empirically.
Revisions of the genetic decomposition may lead to changes in instruction, as
well as provide an opportunity for further empirical analysis. The cycle of instruc-
tion ! analysis ! refinement can be repeated until it is determined that the
refinement adequately describes students’ constructions and leads to effective
instruction.
In their study of the concept of limit, Cottrill et al. (1996) devised a preliminary
genetic decomposition. They based their decomposition on existing mathematics
education literature, their understanding of the concept, and their instructional
Fig. 4.5 Construction of the
Process of linear
transformation (Roa-Fuentes
and Oktac 2010, p. 106)
44 4 Genetic Decomposition
experience. Students completed an instructional sequence that was informed by the
decomposition. An analysis of data gathered from these students revealed the need
for a refinement.
The preliminary genetic decomposition, its refinement, and the rationale for
changes to the preliminary decomposition are given in Table 4.1. The preliminary
decomposition is divided into six steps. The first three describe mental
constructions involved in the development of an informal understanding of the
concept, and the last three involve mental constructions associated with develop-
ment of a formal understanding. Generally speaking, an informal understanding
involves a dynamic conception, that is, the values of a function approach a limiting
value as the values in the domain approach some quantity. A formal understanding
is typically identified with the ε� δ definition.An informal understanding is initially static. For a function f at a domain point
x ¼ a, determination of the limit starts with the Action of evaluating f at a few
points, each successively closer to x ¼ a (Step 1P). As the individual reflects on
these Actions, they may be interiorized into a mental Process (Step 2P); it is at this
point that the static conception becomes a dynamic conception. Encapsulation
occurs as the individual sees the need to apply Actions to the dynamic Process
(Step 3P). The transition to more formal thinking starts with Step 4, as the Process
constructed in Step 2P is reconstructed in terms of intervals. The formal definition
then arises through application of a two-level quantification Schema (Step 5P) to
the reconstructed Process (Step 4P).
The instructional treatment consisted of a 2-week unit that included five types of
computer activities embedded in the usual topics of approximation, one- and
two-sided limits, and applications of the limit. The students conducted graphical
analyses, wrote and analyzed short programs related to the informal, dynamic
conception, and completed tasks involving construction and analysis of ε� δintervals.
The data analysis suggested two major revisions. The first was addition of a
step that precedes the Action of evaluating a function at several points (Step 1R);
specifically, the individual evaluates a single point, which may be x ¼ a itself
instead of a series of selected points that are successively closer to x ¼ a. Thesecond deals with the construction of the Process conception. Rather than a
single Process, the researchers uncovered evidence of a coordination of two
Processes: a domain Process, in which x approaches a, and a range process, in
which y approaches L . The two Processes are then coordinated through
the function f . In other words, the function f is applied to the Process of xapproaching a to obtain the Process of f ðxÞ approaching L. According to Cottrill
et al. (1996), one source of students’ difficulties with the limit concept can be
attributed to this more complicated mental construction. The other principal
source of difficulty, which is related to the formal concept of limit, is that
students do not possess a sufficiently powerful conception of quantification
(Dubinsky et al. 1988).
4.5 Refinement of a Genetic Decomposition 45
Table
4.1
Preliminarygenetic
decompositionoftheconceptoflimitanditsrefinem
ent
Preliminarygenetic
decomposition
Refinem
ent
Rationaleforchange
1R:TheActionofevaluatingthefunctionfat
asingle
pointxthat
isconsidered
tobe
close
to,oreven
equal
to,a
Anumberofstudentsevaluated
asinglepoint,often
isolatingtheirattentiononthepointx¼
a
1P:TheActionofevaluatingthefunctionf
atafew
points,each
successivepoint
closerto
a
2R:TheActionofevaluatingthefunctionf
atafew
points,each
successivepointcloser
toa
2P:InteriorizationoftheActionofStep1P
toasingle
Process
inwhichfðx
Þapproaches
Las
xapproaches
a
3R:Constructionofacoordinated
Process
Schem
a:
(a)InteriorizationoftheActionofStep2R
toconstruct
adomainProcess
inwhichx
approaches
a(b)ConstructionofarangeProcess
inwhich
yapproaches
L(c)Coordinationof(a)and(b)via
f
Thedatashowed
numerousinstancesofthecon-
structionoftwoseparateProcesses.Some
studentsonly
constructed
oneofthem
andthis
appearedto
preventthem
from
understanding
thelimitconcept.Studentswhoweremore
successfulgaveevidence
ofhavingcoordinated
theseProcesses
3P:EncapsulationoftheProcess
ofStep2Pso
that
thelimitbecomes
anObject
towhich
Actionscanbeapplied
4R:EncapsulationoftheProcess
ofStep3R(c)
sothat
thelimitbecomes
anObject
towhich
Actionscanbeapplied
4P:ReconstructionoftheProcess
ofStep2Pin
term
sofintervalsandinequalities.Thisis
donebyintroducingnumerical
estimates
of
theclosenessapproach:0<
x�a
jj<
δand
0<
fðxÞ�
Lj
j<ε
5R:ReconstructionoftheProcess
ofStep3R(c)
interm
sofintervalsandinequalities.Thisis
donebyintroducingnumerical
estimates
of
theclosenessapproach:0<
x�a
jj<
δand
0<
fðxÞ�
Lj
j<ε
Thereconstructed
Process
described
inthisstep
would
bebased
onthecoordinated
Process
rather
than
asingle
Process,as
described
inthe
preliminarygenetic
decomposition
5P:Applicationofatwo-level
quantification
Schem
ato
connecttheProcess
described
in
Step4Pto
theform
aldefinition
6R:Applicationofatwo-level
quantification
Schem
ato
connecttheProcess
described
in
Step5Rto
theform
aldefinition
6P:Applicationofacompletedε�δ
conceptionto
specificsituations
7P:Applicationofacompletedε�δconception
tospecificsituations
46 4 Genetic Decomposition
Step 1R was not part of the preliminary decomposition; analysis of the data
revealed this as a step that precedes Step 2R, which was the first step (Step 1P) in
the preliminary decomposition.
The coordinated Process (Step 3R) replaced the single Process (Step 2P) in
the preliminary genetic decomposition. In the data analysis, students tended to
construct a separate Process for approaching x ¼ a apart from application of the
function f . After constructing the domain Process, students applied the function fto the domain elements constructed through the domain Process to come up with a
range Process.
The coordination of two Processes, one for approaching x ¼ a and the other for yapproaching L through f , shows that the dynamic conception of the limit turned out
to be more complicated than first thought. Unlike some researchers, who believe
that a dynamic conception hinders progress toward development of formal under-
standing, Cottrill et al. (1996) found that students’ difficulty can be attributed, at
least in part, to an insufficiently well-developed dynamic conception, which
appears to need to be based on a coordinated Process Schema.
4.6 Role of the Genetic Decomposition in the Design
of Teaching Activities
In addition to being a theoretical model for research, the genetic decomposition for
a concept guides instruction. Since a genetic decomposition describes the
constructions a student may need to make in order to learn a mathematical concept,
it can be used to design activities to help students to make the proposed
constructions. Although going from the genetic decomposition to the design of
instructional activities is not always direct, the way in which the former informs the
latter is very important since it represents a bridge between the theory and its
pedagogical use (Trigueros and Oktac 2005).
In a teaching sequence whose design is based on APOS theory, the first part of
the teaching cycle, which is described in detail in Chap. 5, consists of activities
for students to work on collaboratively, often in a laboratory setting. Each of the
activities is designed to provide opportunities for students to repeat specific Actions
and to reflect on them, to foster interiorization of Actions into Processes, to help
with the coordination and reversal of Processes, and to support encapsulation of
Processes into Objects. A teaching sequence can also include activities where the
goal is the construction of relationships among different Actions, Processes,
Objects, and previously constructed Schemas. These activities may help students
construct a new Schema or, in the case of a previously constructed Schema, lead to
further development or refinement of that Schema.
Research can also be conducted following instruction. The focus of the
research is to determine whether students made the constructions predicted by
the genetic decomposition and whether such constructions helped them to learn
the mathematics in question. The data that is obtained is compared with what is
4.6 Role of the Genetic Decomposition in the Design of Teaching Activities 47
predicted by the genetic decomposition. This type of analysis may lead to refine-
ment of the genetic decomposition. When a genetic decomposition accurately
reflects the mental constructions students make in their efforts to understand a
concept, comparative research can be conducted to compare the mathematical
performance of students who have completed APOS-based instruction with
students who have completed instruction that was not APOS based. Research of
this type has been conducted and has shown the promise of instruction based on
APOS Theory (Weller et al. 2003).
In order to further illustrate how a genetic decomposition informs the design
of activities, examples from the vector space chapter from the textbook
Learning Linear Algebra with ISETL (Weller et al. 2002) are presented in
Sects. 4.6.1 and 4.6.2.
4.6.1 Genetic Decomposition of a Vector Space
The concept of vector space is a Schema that is constructed by coordinating the three
Schemas of set, binary operation, and axiom. The set and binary operation Schemas
are thematized to form Objects and coordinated through the axiom Schema.
Binary Operation. A binary operation is a function of two variables defined on
a single set or on a Cartesian product of two sets. In terms of the APOS framework,
there are four mental constructions involved in development of this concept:
Action: Given a formula for a binary operation, an individual can take two
specific elements of the set(s) on which the operation is defined and
apply the formula.
Process: The individual interiorizes the Actions comprising a binary
operation: accepting two elements, acting on these Objects in some
way, and returning a new Object.
Object: The individual can distinguish between two binary operations,
consider more than one binary operation defined on a set or on a
product of sets, check whether a binary operation satisfies an axiom,
and de-encapsulate a binary operation so that it can be coordinated
with other Processes.
Schema: The individual can define a binary operation on a set or on a product
of two sets and/or identify whether a function defined on a set, or sets,
is a binary operation.
Set. A set is a collection of Objects that satisfies a given condition. In terms of
the APOS framework, there are four mental constructions involved in development
of this concept:
Action: An individual can only conceive of a set when given a specific listing
of elements or when presented with a particular condition of set
membership.
48 4 Genetic Decomposition
Process: The Action of gathering and putting Objects together in a collection
according to some condition is interiorized.
Object: The individual can apply Actions or Processes to the Process such as
determine the cardinality of a set, compare two sets (not necessarily
in terms of cardinality), consider a set to be an element of another set,
and define a function in which a set is one of the Objects accepted.
The student can also de-encapsulate a set so that it can be coordinated
with other Processes.
Schema: The individual can apply the set Schema to a given mathematical
situation. In linear algebra, this would mean being able to define
sets of Objects that might later be classified as sets of vectors:
tuples, polynomials, functions, and matrices. An individual may
have also developed a general notion of what a set is and what
it is not.
Axiom. For vector space, an axiom is a Boolean-valued function that accepts a
set, or a Cartesian product of sets, and a binary operation defined on the set, or sets,
and checks whether the property defined by the axiom is satisfied. Checking an
axiom is a Process that involves coordination of the general notion of checking a
property with the Process defined by the specific property being checked. When an
axiom is applied to a set and a binary operation, the set and binary operation
Schemas must be de-encapsulated and coordinated with the Process of checking
the property in question. An axiom Schema includes the general notion of checking
whether a set, operation pair, satisfies a property.
How the Three Schemas Work Together in the Mental Construction of a
Vector Space. The axiom Schema includes the general notion that a binary
operation on a set may or may not satisfy a property and the ten specific Objects
obtained by encapsulating the ten Processes corresponding to the ten vector space
axioms. Each axiom is de-encapsulated for individual coordinations to take place.
Each coordinated Process is applied to the set and binary operation Schema. The set
and binary operation Objects are de-encapsulated so that they can be coordinated
with each axiom. The ten instances of this operation are then coordinated into a
single Process of satisfying the axioms. The interaction of these Schemas is
illustrated in Fig. 4.6.
4.6.2 Activities Designed to Facilitate Developmentof the Vector Space Schema
Unlike many linear algebra texts that start with systems of equations and matrices,
Learning Linear Algebra with ISETL begins with vector spaces. This chapter
is preceded by an introductory chapter on functions and structures. Although one
4.6 Role of the Genetic Decomposition in the Design of Teaching Activities 49
of the purposes of the introduction is to familiarize students with ISETL commands,
students learn about these commands in the context of working with mathematical
objects such as sets. Students study different representations for sets and convert
from one representation to another (say, from a description of a set to a set
former or from a set former to a list); they also construct and compare sets and
distinguish sets from other objects such as tuples. The objective is development of a
set Schema.
In both chapters (the introduction and the chapter on vector spaces), students
interpret and write ISETL code for a variety of funcs. A func is an ISETL
command for a function. A func accepts variable(s), whose values can be num-
bers, sets, tuples, and even other funcs. It includes a return statement that yields
the output of the func. In two of the activities, students construct funcs to carry
out the binary operations of addition (modulo p) and multiplication (modulo p) on afinite field (i.e., Zp). They use these funcs in the construction of other funcs toperform addition and scalar multiplication on sets of tuples Zn
p (n a positive integer)
over the field Zp.Once students have constructed funcs for tuple addition and scalar
multiplication for different pairs of scalar fields and sets of tuples, they begin to
write code to test properties of the operations they have defined; specifically, to test
the vector space axioms. Once they have written the code for a particular axiom,
they test it on different systems ðK; V; va; smÞ4 they have defined and worked
with in previous activities. Interpretation of code involving the vector space axioms
Fig. 4.6 Diagrammatic representation of the genetic decomposition of the vector space Schema
4Here, K refers to a field, V stands for a set of tuples, va denotes addition defined on V, and sm
represents the scalar multiplication operation defined on K and V.
50 4 Genetic Decomposition
encourages interiorization of the Process underlying each axiom. Applying a funcfor an axiom facilitates encapsulation of the set and binary operation concepts since
they are accepted as inputs to the func. Writing a func for an axiom supports
development of the axiom Schema since the individual needs to coordinate the
Process of checking a property with the Process associated with the specific axiom
being worked with.
Eventually, students are presented with all ten axioms and asked to explain
how each axiom works. They then apply the axioms to 12 different systems
ðZp; Znp ; va; smÞ (for different values of p and n). They generate a table in
which they record, for each system, which axioms are satisfied. In the subsequent
activity, they summarize their findings. The section culminates with construction of
the funcis_vector_space, a Boolean-valued function that accepts a set V, afield of scalars K, an operation va defined on V, and an operation sm defined on the
pair ðK;VÞ . The func tests whether the system ðK;V; va; smÞ satisfies all ten
axioms. It returns true if all ten axioms are satisfied and false otherwise. The
purpose of having students write and use this func is to support the mental
constructions called for by the preliminary analysis and to coordinate the ten
Processes underlying each axiom into a single Process that establishes whether
the system constitutes a vector space.
4.7 What Is Not a Genetic Decomposition
By now it may be quite clear what a genetic decomposition is. However, given its
complexity, in some research projects and papers and also in students’ work, it
happens that what authors or students call a genetic decomposition is not really one.
In what follows, some examples, stated verbatim, of such “genetic decompositions”
are presented and discussed.
It is a common error to confound a genetic decomposition with a description of a
teaching sequence or a mathematical description of a concept where APOS termi-
nology is used. Example 1 on matrices, offered by a teacher who participated in a
seminar on APOS Theory, illustrates this:
Example 1 of what a genetic decomposition is not:
1. Students do Actions to define a matrix.
2. Students do Actions to define size of matrices.
3. Process for adding two matrices.
4. Process to multiply matrices by scalars.
5. Processes to multiply two matrices with restrictions on this operation.
6. Processes to verify properties of operations.
7. Actions to verify if the inverse of a matrix exists.
8. Process to find the inverse of a matrix.
9. Encapsulation of the concept of matrix.
4.7 What Is Not a Genetic Decomposition 51
For Example 1, instead of offering a description of specific mental constructions
needed to learn the concept of matrix, this teacher described a class plan that
consisted of a list of mathematical topics to cover. Although APOS terminology
was used, it was not related to the cognitive structures of APOS Theory, that is,
what was proposed as a genetic decomposition does not specify the Objects on
which Actions or Processes are performed, does not include any description of how
Actions are interiorized into Processes, nor describes how Processes are
encapsulated or coordinated. Moreover the steps do not even identify what the
Actions, Processes, or Objects are.
Another common error is to describe a “genetic decomposition” that simply lists
operations a student is to perform. This is exemplified in Example 2 on trans-
formations of real-valued functions presented by a graduate student in a seminar:
Example 2 of what a genetic decomposition is not:
Action: Can see specific examples as representing a transformation of a known
function. Can draw the graphs of translations of a real-valued function when the
graph of the function is well known, such as a linear or quadratic function. Can
introduce values into the rule of a transformed function to obtain its value. Can
find the graph of a transformed function using points
Process: Can draw general basic transformations of a given function (translate it
vertically or horizontally, stretch it). Can find transformations of a given func-
tion. Can determine the original function if given a certain transformed function.
Understands the difference between horizontal and vertical translations, and of
stretching functions in general, and sees if translations modify the domain and
range of the original function
Object: Can operate on transformed functions to obtain new functions such
as the composition of transformations. Can draw the graph of any transformed
function. Can relate any transformed function to the original function. Can
predict the function that results from a composition of transformations.
Understands the difference between diverse transformations of functions in
general and how the domain and range of a function change when the function
is transformed
Although the terms Action, Process, and Object are included in the graduate
student’s proposed “genetic decomposition,” they are stated in terms of students’
conceptions, that is, they are not stated in terms of the constructions a student
needs to make in order to perform the activities listed. For example the description
does not specify the Actions involved in graphing transformations point by point,
explain how those Actions are interiorized so that students can recognize
transformations graphically, or tell how to reverse the Process involved in
recognizing the original function when given a transformed function.
Some of the difficulties encountered in the design of a genetic decomposition
can be related to misunderstandings of the theory that are reflected in the way the
constructions are described. This can be observed in Example 3 on the derivative
that was presented by a graduate student in a seminar:
52 4 Genetic Decomposition
Example 3 of what a genetic decomposition is not
Preliminary Knowledge
Geometric Rate as an Object. The students need to have assimilated it, be
aware of the meaning of the rate as a totality and be able to do Actions on it, and
see it as a trigonometric rate: the tangent of a linear function.
Secant as an Action. The student must have assimilated it so that he or she is
able to manipulate it physically or mentally.
Tangent to a Circle. It must be assimilated as a Process because the student
must be capable of coordinating different definitions of tangents to a circle.
Linear Function. Given the slope and a point, determine the equation of a
linear function and think of it as a coherent collection of Objects (slope and
point), Actions (on those Objects), and Processes (manipulations to find the
equation of a line and its graph), so it must be a Schema. Functions: R!R.The student should have developed a Schema for functions as a collection of
Objects (graphs and algebraic expressions), Actions (on the Cartesian plane
and algebraic expressions), and Processes (manipulation of algebraic
expressions together with Cartesian plane to draw the graph).
Limit. Process of approaching closer and closer.
The following comments can be made about the preliminary knowledge in
Example 3. In the paragraph Secant as an Action, the author is not aware that if
the student can do manipulations mentally, it means that he or she has interiorized
the Actions into a Process. In addition, nothing is said about the Actions that are
included in such manipulations. In Tangent to a circle, the author applies the notionof coordination to definitions instead of to Processes. In Linear function the
conception of Schema is not clear in the last two points.
Continuation of Example 3
“Genetic decomposition” of derivative
1. The geometric rate as an Object, the function for which the derivative has to
be obtained as a Schema, and the secant to a circle as an Action are coordi-
nated in a new Schema to construct the secant of a function which must be
considered as a new Object, that is, it has to be thematized into a Schema.
2. The Action limit is applied to the Object secant.
3. When the tangent to a point A of the function is interiorized into a simple
Process to construct the coefficient that determines the slope of the tangent
line to the function on the point A.
4. A Schema is created with the Object tangent to the point A of the function and
the Schema linear function: a coherent collection of Objects (slope and point),
Actions (on the slope and point), and Processes (manipulation to find the linear
function and to draw its graph).
4.7 What Is Not a Genetic Decomposition 53
5. The Processes of the Schema in 4 are coordinated to the interiorized Action in
3 to create the Process, derivative of the function in the point A of the domain
of the function.
6. All the Processes of the derivatives at each point of the domain of the function
are coordinated to encapsulate them in the Schema derivative function of
function f.7. The Schemas for limit of a secant line when the denominator is approaching
zero, equation of the derivative of a function, and that of graphical represen-
tation of the derivative function.
The problem in Example 3 is that the meaning of the constructions does not
appear to be understood; for example, a “simple” Process is mentioned without
reference to the Actions from which it arises; a coordination of Actions with
Processes is mentioned, but it is not clear what is meant by “the Process of the
Schema” in number 3; the meaning of number 5 is not clear; a different Process for
each of the points of the function is considered in number 6 when this should be the
interiorization of Actions; in the same item, there is no mention of the Actions that
led to encapsulation of the Process; there is a misuse of the term Schema; and
finally, it is very difficult to make sense of what the student wrote in number 7.
In some cases, a “genetic decomposition” consists of a description of what
researchers consider an Action, Process, and Object conception of the concept, as
is exemplified in Example 4, taken from a final presentation of a student in a
graduate program.
Example 4 of what a genetic decomposition is not:
If a student has an Action conception, he or she is limited to do Actions. The
Actions that the student shows are:
A.1. Has memorized that three noncoplanar vectors in R3 is a basis for this space.
A.2. Can find a basis for a subspace of R3 by manipulating a given equation, for
example, the equation of a plane or a line.
A.3. Can perform Actions on a given set of vectors to verify if they are or not
linearly independent.
A.4. Can verify that given sets of three vectors span or not R3.
The Process conception of a student is demonstrated by her or his possibility to
show that he or she has interiorized these Processes:
P.1. Can find a basis for any vector space.
P.2. Can verify the linear independence of any set of vectors given.
P.3. Can verify if the vectors of a given set span or not a given vector space.
P.4. Demonstrate that he or she has not coordinated the previous Processes and
has not encapsulated basis as an Object because he or she has difficulties to
distinguish spanning sets from basis.
The Object conception of a student is demonstrated by
O.1. Can perform operations on any given basis
54 4 Genetic Decomposition
O.2. Can compare and distinguish different sets and decide if they are or not a
basis for a given vector space
O.3. Can consider and find basis for infinite spaces such as Rn
O.4. Can use the concept of basis for vector spaces different from Rn
From a methodological point of view, Example 4 cannot be considered a genetic
decomposition since the mental constructions needed to learn the concept are not
described. For example, there is no description of the Actions that are interiorized
into the Process of finding a basis for any vector space, nor an explanation of the
result of coordination of Processes P2 and P3.
While these examples of erroneous “genetic decompositions” include APOS
terminology, each falls short in some fundamental respect. What is important to
learn from these examples is that a genetic decomposition is far more than a
sequence of steps for instruction or a list of conceptions students may have. Rather,
it is a description of the mental constructions students may need to make in their
learning of a mathematical concept. In this sense, a genetic decomposition is a
guide for the design of instruction that aligns with how students come to understand
a mathematical concept.
4.7 What Is Not a Genetic Decomposition 55
Chapter 5
The Teaching of Mathematics
Using APOS Theory
This chapter is a discussion of the design and implementation of instruction using
APOS Theory. For a particular mathematical concept, this typically begins with a
genetic decomposition, a description of the mental constructions an individual
might make in coming to understand the concept (see Chap. 4 for more details).
Implementation is usually carried out using the ACE Teaching Cycle, an instruc-
tional approach that supports development of the mental constructions called for by
the genetic decomposition.
The ACE cycle includes activities, which students typically work on coopera-
tively, sometimes with use of a mathematical programming language such as the
Interactive Set Theoretic Language (ISETL). The phrase mathematical program-
ming language refers to a program that satisfies three properties:
1. The syntax is close to standard mathematical notation.
2. Certain mathematical features are supported together with their usual mathemat-
ical properties.
3. Important data types, such as procedures and functions, can be operated on and
called and returned by procedures and functions.
The components of the ACE Teaching Cycle and the features of ISETL are
described in Sects. 5.1 and 5.2. The remainder of the chapter is devoted to examples
of APOS-based instruction on groups in abstract algebra for mathematics majors
(Sect. 5.3) and on infinite repeating decimals for a content course for preservice1
elementary and middle school teachers (Sect. 5.4).
1 Preservice refers to college or university students who are preparing to become school teachers.
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_5,© Springer Science+Business Media New York 2014
57
5.1 The ACE Teaching Cycle
The ACE Teaching Cycle is a pedagogical strategy that consists of three
components: (A) Activities; (C) Classroom Discussion; and (E) Exercises.
For Activities, which constitute the first step of the cycle, students work cooper-
atively in teams on tasks designed to help them to make the mental constructions
suggested by the genetic decomposition. The focus of these tasks is to promote
reflective abstraction rather than to obtain correct answers. This is often achieved
by having students write short computer programs using a mathematical program-
ming language.
The Classroom Discussion, the second part of the cycle, involves small group
and instructor-led class discussion, as students work on paper and pencil tasks that
build on the lab activities completed in the Activities phase and calculations
assigned by the instructor. The class discussions and in-class work give students
an opportunity to reflect on their work, particularly the activities done in the lab. As
the instructor guides the discussion, he or she may provide definitions, offer
explanations, and/or present an overview to tie together what the students have
been thinking about and working on.
Homework exercises, the third part of the cycle, consist of fairly standard
problems designed to reinforce the computer activities and the classroom discus-
sion. The exercises help to support continued development of the mental
constructions suggested by the genetic decomposition. They also guide students
to apply what they have learned and to consider related mathematical ideas.
The ACE Cycle and its relationship to the genetic decomposition are illustrated
in Fig. 5.1.
The arrow from the Genetic Decomposition to the dotted box illustrates the fact
that the genetic decomposition affects each component of the ACE Teaching Cycle.
The bidirectional arrow between Activities and Classroom Discussion shows
that, on the one hand, the activities are the principal subject of the class discussion
Fig. 5.1 Relation between the ACE Teaching Cycle and a genetic decomposition
58 5 The Teaching of Mathematics Using APOS Theory
and, on the other, that the classroom discussion provides an opportunity for the
students to reflect on the activities. The arrows to Exercises from Activities and
Classroom Discussion reflect the principal purpose of the exercises—to reinforce
the mental constructions the students make or have begun to make as they work
through the Activities and participate in the Classroom Discussion.
The Activities phase involves completion of cooperative tasks informed by
the genetic decomposition. Although computers have frequently been involved,
their use is not required. It is simply the case that activities involving use of a
mathematical programming language have been effective in helping students in
learning a mathematical concept using the mental constructions called for by a
genetic decomposition for the concept (seeWeller et al. 2003). ISETL, the language
typically used, is described in the next section.
5.2 ISETL: A Mathematical Programming Language
5.2.1 A Brief Introduction to ISETL
ISETL is a freeware mathematical programming language. What separates ISETL
from other programming languages is its ability to represent mathematical concepts
using mathematical notation and the ability of the language to operate on the
concepts represented by that notation. The program can be obtained online from
one of the following URLs:
http://titanium.mountunion.edu/isetlj/isetlj.htmlhttp://homepages.ohiodominican.edu/~cottrilj/datastore/isetl/;
Dautermann (1992) wrote a manual that provides details regarding the use of
ISETL, including its commands and features. The reader may wish to download and
use ISETL to work through the examples that appear in this section.
ISETL has proven to be a powerful tool in helping students to learn mathematics.
The syntax of the language is very close to standard mathematical notation, the
language supports certain mathematical features, and all data types can be acted
on as objects. Each of these aspects of ISETL is discussed in Sects. 5.2.2–5.2.4.
Use of ISETL as a pedagogical tool is considered in Sect. 5.2.5.
5.2.2 The Syntax Is Close to Standard MathematicalNotation
In ISETL, syntax resembles standard mathematical notation. For example, in
mathematical set former notation, the prime numbers from 2 to 100 can be
represented in the following way:
5.2 ISETL: A Mathematical Programming Language 59
fx : x 2 2; 3; . . . ; 100f g j ð69 y 2 2; 3; . . . ; x� 1f g 3 x mod y ¼ 0g:
The ISETL representation is nearly identical:
{x : x in {2..100} | (not exists y in {2,3..x-1} | x mod y¼0}.
In ISETL the word “in” stands for the “element of” symbol 2 , the phrase
“not exists” replaces the “not exists” symbol 6 9, and the character “|” represents
the “such that” symbol 3 .
To return a set in list form, the user types the code for the set former at the ISETL
prompt >, places a semicolon at the end of the code, and then presses enter. For the
set of prime numbers less than 100, the screen display for returning the set as a list is
> !setrandom off> {x : x in {2,3..100} | not exists y in {2,3..x-1} | x mod
y ¼ 0};{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47,
53, 59, 61, 67, 71, 73, 79, 83, 89, 97};or> !setrandom on> {x : x in {2,3..100} | not exists y in {2,3..x-1} | x mod
y ¼ 0};{17, 19, 29, 23, 11, 13, 7, 5, 2, 3, 41, 37, 31, 47, 43, 53,
59, 61, 67, 71, 73, 97, 89, 83, 79};
The command !setrandom off ensures that the elements of the set will be
listed in numerical order, while the command !setrandom on returns the listing
of the elements in random order.
The syntax for sets of tuples is treated in a similar way. In standard mathematical
notation, the set-builder representation for Z32, the set of all three tuples with entries
in Z2 (the set of all integers modulo 2), is
Z32 ¼ a; b; cð Þ : a; b; c 2 0; 1f gf g:
In ISETL, the set is represented almost identically:
Z2_3:¼{[a,b,c] : a,b,c in {0,1}};
There is, however, a subtle, though important, difference between the two
representations. In the mathematical representation, the ordered triples appear
with parentheses. In ISETL, the ordered triples appear with square brackets. The
reason is that ordered sequences in ISETL are expressed using bracket [ ] notation.
When using mathematical notation, the set Z32 in list form appears as
Z32 ¼ f 1; 1; 1ð Þ; 1; 0; 1ð Þ; 1; 0; 0ð Þ; ð0; 1; 0Þ; 0; 1; 1ð Þ; 0; 0; 1ð Þ; 0; 0; 0ð Þ; ð1; 1; 0Þg;
60 5 The Teaching of Mathematics Using APOS Theory
while, in ISETL, it is represented as
Z2_3:¼{[1,1,1],[1,0,1],[1,0,0],[0,1,0],[0,0,1],[0,0,0],[1,1,0]}.
Numbers and operations resemble scientific or graphing calculator displays.
The only difference is the prompt given by the symbol >. Users enter code on
lines that begin with this symbol or with the symbol>>, which indicates input to be
completed. Completion of input is typically indicated with a semicolon. For
example, in Line 4 following the entry 13*(233.8), the program prompt>> appears
because no semicolon was added when the expression was entered. Once a semico-
lon is entered, the result of the computation is returned (Line 5). Lines without
prompts indicate what the computer returns as output2:
Line 1: > 7+18;Line 2: 25;Line 3: > 13*(233.8)Line 4: >> ;Line 5: 3039.400;Line 6: > !rational offLine 7: > 27/36;Line 8: 0.750;Line 9: > !rational onLine 10: > 27/36;Line 11: 3/4;Line 12: > 5-9Line 13: >> ;Line 14: -4;
The command !rational off (Line 6) instructs ISETL to return rational
numbers in decimal form (the number of decimal places can be set using a simple
command), while the command !rational on (Line 9) calls for fractional
representations.
ISETL supports variables, which are case sensitive and defined using the
symbol:¼. For example, entry of x:¼2; sets the variable x equal to 2. When
the variable x is called (indicated in ISETL by x;), ISETL returns 2:
> x:¼2;> x;2;
2 The notation Line # does not actually appear on an ISETL screen. It is being used for convenience
here and in other examples with longer lines of code.
5.2 ISETL: A Mathematical Programming Language 61
In addition to variables, ISETL supports Boolean operators (and, or, not,and impl, where impl is shorthand for implies), the usual comparison operators
(¼, /¼ (inequality), <, <¼, >, >¼), and quantifiers (forall, exists,not exists).
5.2.3 Supporting Mathematical Features
Certain control statements such as if statements and for loops can be used to
carry out mathematical procedures without the need to invoke special commands.
In Line 1 of the following example, the variable x is set equal to 4. For the ifcommand, which appears in Lines 2 through 6, the beginning statement is of
the form if [condition] then and ends with the line end if;. Becausethe condition x> 2 is satisfied and precedes the condition x> 3 in Line 4 where the
elseif statement appears, ISETL only returns the first phrase, x is largerthan 2. For a value of x less than or equal to 2, ISETL would not return anything:
Line 1: > x:¼4;Line 2: > if x>2 thenLine 3: >> writeln "x is larger than 2";Line 4: >> elseif x>3 thenLine 5: >> writeln "x is larger than 3";Line 6: >> end if;Line 7: x is larger than 2
Like an if statement, a for loop has both a beginning and a concluding
statement. The beginning statement is of the form for [element(s) insome set] do, and the concluding statement is given by the end for command.
In the sample code below, Lines 1 and 2 define variables that will be used in
execution of the for loop: the set S, which represents the first three counting
numbers, and the variable a, which is set equal to 0 (as an initial value). The forloop, given in Lines 3 through 5, builds a sum, represented by the variable
a, consisting of all possible pairs of elements from S. The value of a is reported
in Line 7 as a result of the ISETL “call” for the value of the variable, which appears
in Line 6.
Line 1: > S:¼{1..3};Line 2: > a:¼0;Line 3: > for x, y in S doLine 4: >> a:¼a+x+y;Line 5: >> end for;Line 6: > a;Line 7: 36;
Important mathematical objects such as functions can be defined and evaluated
in ISETL. In the following case, the command func (Line 1) accepts variable(s),
62 5 The Teaching of Mathematics Using APOS Theory
whose values can be numbers, sets, tuples, and even other funcs. It includes a
return statement (Line 2) that yields the output of the func. The code representsa function f that accepts a number x and returns the sum of x and 3 modulo 6. Line
5 gives the code necessary to evaluate f when x ¼ 4. The actual value of f at x ¼ 4
is given in Line 6.
Line 1: > f:¼func(x);Line 2: >> return (x+3) mod 6;Line 3: >> end;Line 4: >Line 5: > f(4);Line 6: 1;
A proc or procedure is the same as a func except that it has no return
statement and does not return a value. It is used to perform internal operations.
In the example below, the proc called SetNot accepts a tuple of length 2 called
pair. The variable G, the first component of the tuple (a set, defined in Line 7), is
assigned the value of pair(1), and the variable o, the second component of
the tuple (a func, defined in Lines 8–11), is assigned the value of pair(2).SetNot assigns to the variable e the identity with respect to o and G. The value
of the variable inv is a set that consists of all the tuples of the form [g, g’],where g is an element of G and g’ is the left inverse of g with respect to the
operation o. The “.o” notation that appears in Lines 3 and 4 in the equations x .og¼ g and g’ .o g¼ e indicates use of .o as a binary operation on G in infix form.
The infix notation can be used with any func of two variables in ISETL. If o is a
func that accepts two inputs a and b, the call for o can be expressed as o(a,b) or
as a .o b. This latter notation acknowledges that a function of two variables
defined on a set is a binary operation on that set:
Line 1: > SetNot:¼proc(pair);Line 2: >> G:¼pair(1); o:¼pair(2);Line 3: >> e:¼choose x in G | (forall g in G | x .o g ¼ g);Line 4: >> inv:¼{[g, choose g’ in G | g’ .o g¼ e] : g in G};Line 5: >> end;Line 6: > pair:¼[ ];Line 7: > pair(1):¼{0..5};Line 8: > pair(2):¼func(x,y);Line 9: >> if (x in G and y in G) thenLine 10: >> return (x*y) mod 6;Line 11: >> end;Line 12: >> end;Line 13: > pair;Line 14: [{0, 1, 2, 3, 4, 5}, !func(6)!];Line 15: > SetNot(pair);Line 16: > G; e; inv(5); inv(2);Line 17: {1, 0, 2, 3, 5, 4};
5.2 ISETL: A Mathematical Programming Language 63
Line 18: 1;Line 19: 5;Line 20: OM;
In Line 13, where pair appears, ISETL returns a tuple which consists of two
elements, the set defined by pair(1) and an expression !func(6)!, whichdenotes the fact that pair(2) is a func.
Line 15 denotes the “call” of SetNot for pair. Nothing is returned because
the proc merely performs the operations given in its lines of code. Results of the
execution of SetNot(pair) are given in Lines 16–20. Line 16 calls for return of
the set G, the element e, and the inverses of 5 and 2, mod 6. Line 17 yields the set Gin list form, Line 18 returns the identity element of G with respect to the operation
o, Line 19 returns the inverse of 5 with respect to the operation o, and Line
20 returns OM to reflect the fact that 2 has no inverse with respect to the operation o.
5.2.4 Operations on Data Types
ISETL is powerful in part because it can support operations on data types. From a
technical point of view, this means that certain ISETL data types are first-class
objects. Funcs represent one of the most important examples. For instance, a funccan actually return another func. This can be seen in the following lines of code
for the func D, which accepts a function f and returns a func, which is the
difference quotient of f within 0.001 units of the value of the variable x:
> D:¼func(f);>> return func(x);>> return (f(x+0.001)-f(x))/0.001;>> end;>> end;
The func D enables ISETL to compute the difference quotient for any specified
function f at any domain point x using a difference of 0.001. For instance, if the
function f is defined by f ðxÞ ¼ x2 þ 1, which would appear in ISETL as
> f:¼func(x);>> return x**2+1;>> end;
the call of D(f)(3); returns the value of the difference quotient for f at x ¼ 3
using a difference of 0.001:
> D(f)(3);6.001;
A similar example is the func Sum, given by the code which follows. This
func accepts two funcs representing functions f and g and returns a functhat represents their sum, which is given by Sum, which is defined in Lines 1–5.
64 5 The Teaching of Mathematics Using APOS Theory
Two particular functions f and g are defined in Lines 7–13. The call for Sum, givenby Sum(f,g) (Line 15), returns the value !func(7)! (Line 16) to indicate that
Sum(f,g)is itself a func. The call for Sum(f,g)(2) (Line 18) yields the value
of f þ gð Þð2Þ (Line 19):Line 1: > Sum:¼func(f,g);Line 2: >> return func(x);Line 3: >> return f(x)+g(x);Line 4: >> end;Line 5: >> end;Line 6: >Line 7: > f:¼func(x);Line 8: >> return x**3+2;Line 9: >> end;Line 10: >Line 11: > g:¼func(x);Line 12: >> return 3*x;Line 13: >> end;Line 14: >Line 15: > Sum(f,g);Line 16: !func(7)!;Line 17: >Line 18: > Sum(f,g)(2);Line 19: 16;
5.2.5 ISETL as a Pedagogical Tool
At a functional analysis conference in 1969, Dubinsky (1995) first heard about the
programming language SETL, the forerunner to ISETL, from its developer Jack
Schwartz (see Chap. 2 for more details). Schwartz wanted to express complex
mathematical relationships using computer programs. He believed the best way to
achieve this goal was to base a programming language on fundamental mathemati-
cal concepts. This idea served as the inspiration for the development of APOS
Theory and its related research framework. Specifically, Dubinsky reformulated
Piaget’s ideas about reflective abstraction into a cognitive theory and connected
instruction based on that theory with a pedagogical approach in which students
write short computer programs using a mathematical programming language.
As noted in Asiala et al. (1996),
. . .students gain experience constructing actions corresponding to selected mathematical
concepts. This experience is built upon in subsequent activities where students are asked to
reconstruct familiar actions as general processes. Later activities presented exemplify those
that are intended to help students encapsulate processes to objects; these activities typically
involve writing programs in which the processes to be encapsulated are inputs and/or
outputs to the program. (p. 16)
5.2 ISETL: A Mathematical Programming Language 65
The idea of this pedagogical approach is that computer activities support the
activation of mental mechanisms (i.e., interiorization and encapsulation) that lead
to the development of mental structures (i.e., Processes and Objects) that underlie
the cognitive formation of mathematical concepts. This is illustrated in Fig. 5.2.
Typically, the construction of a new concept that starts as an Action is applied to
an existing physical3 or mental Object (the dotted arrow denotes the fact that the
Action is not part of the Object itself). From an instructional perspective, this
involves computational tasks with explicit instructions and specific examples.
Students construct Actions as they repeat on their computer screens what is written
in the text, predict the result of running code, or modify code they have been given.
This is represented by the “left arm” of Fig. 5.2.
Reflection on an Action leads to interiorization of the Action into a mental
Process. In terms of instruction, interiorization is supported by replacing code that
performs a specific calculation by a short program that carries out the calculation
for unspecified values; that is, the computation is transformed by the learner from a
specific calculation to a general procedure. This is represented by the top half of the
“right arm” of Fig. 5.2.
As an Action is applied to a Process, the Process may be encapsulated into a
cognitive Object (represented by the bottom half of the “right arm” of Fig. 5.2).
Fig. 5.2 Computer-based pedagogical approach, mental mechanisms, and mental structures
3 The application of Actions to physical (real world) Objects is considered in detail in Chap. 9.
66 5 The Teaching of Mathematics Using APOS Theory
From an instructional perspective, encapsulation may occur when a Process is
treated as the input or output of a program, used as a subroutine in a more elaborate
program, or operated on within a program.
Examples of how the computer works as a pedagogical tool within the larger
framework of the ACE Teaching Cycle are considered in detail in Sects. 5.3
and 5.4.
5.3 Teaching and Learning Groups
Abstract algebra is often a mathematics majors’ first encounter with the study of
abstract mathematics. Although many abstract algebra objects include familiar
examples (for instance, the integers as an example of a commutative ring or the
rational numbers as an example of a field), students have not considered these types
of structures in their previous course work. As a result, students often experience
significant frustration when they take an abstract algebra course.
As one response to this problem, Dubinsky and Leron (1994) developed a course
based on APOS Theory that is delivered using the ACE Teaching Cycle and
involves use of ISETL. In this section, examples of instruction on the group concept
will be described.
The genetic decomposition of the concept of group can be understood as a
Schema that consists of three Schemas: set, binary operation, and axiom. The set
and binary operation Schemas are thematized to form Objects and coordinated
through the axiom Schema (Brown et al. 1997). The interaction of these three
Schemas is illustrated in Fig. 5.3.
Fig. 5.3 Diagrammatic representation of the genetic decomposition of the group schema
5.3 Teaching and Learning Groups 67
The axiom Schema includes the general notion that a binary operation on a set
may or may not satisfy a property, which is essentially the Process of checking the
property. It also includes four specific Objects obtained by encapsulating the four
Processes corresponding to the four group axioms. Checking an axiom consists of
coordinating the general notion of checking a property with the specific Process for
the axiom. Each axiom is de-encapsulated so that the individual coordinations can
take place. Each coordinated Process (general property of checking an axiom and
the axiom Process) is applied to the set and binary operation Schema. The set and
binary operation Objects are de-encapsulated so that they can be coordinated with
each axiom. The four instances of this operation are then coordinated into a single
Process of satisfying the axioms (this description is taken from Brown et al. 1997,
p. 192). The way in which this preliminary theoretical description is used to
motivate instruction on groups is considered in Sects. 5.3.1–5.3.3.
5.3.1 Activities
5.3.1.1 Sets
Early activities in the abstract algebra course provide students with opportunities to
develop the concept of set as Process. This occurs as students write set former code
in ISETL or reflect on code provided in the text. For instance, in order to express a set
in set former notation for the subsetH of even elements of Z20 (integers modulo 20),
which is given below, a student would need to carry out in her or his mind the Action
of checking the condition for set membership (determination of whether an element
is even) for every element of the set. This type of activity would likely lead the
student to reflect on the Action of set formation, which would lead to interiorization
of that Action into a mental Process.
> H:¼{g : g in Z20 | even(g)};
A similar type of cognitive activity would be necessary for a student to interpret
a quantification statement such as
> forall x in Z20 | (x+0) mod 20 ¼ x;
In order to predict and then to verify the Boolean value of the quantified
statement, without having to check each element of Z20 explicitly, the student
would need to interiorize the Action associated with checking the condition given
by the universal quantifier for each member of the set.
Dubinsky and Leron (1994) designed activities to support construction of the
concept of set as a mental Object. Each involved the application of Actions on sets.
Two examples follow.
Create New Sets from Old Ones: The students were assigned activities in which
they formed the union and intersection of sets and determined whether one set is a
68 5 The Teaching of Mathematics Using APOS Theory
subset of the other. Sample ISETL code that accompanies these types of activities
is as follows:
Line 1: > Z20:¼{a mod 20 : a in [-30..50]};Line 2: >Line 3: > H:¼{g : g in Z20 : even(g)};Line 4: >Line 5: > K:¼{(5*g) mod 20 : g in Z20};Line 6: >Line 7: > H union K;Line 8: {14, 12, 10, 8, 18, 16, 15, 0, 2, 4, 6, 5};Line 9: > H inter K;Line 10:{0, 10};Line 11: > H subset K;Line 12: false;
For the three sets given in Lines 1, 3, and 5 above, students predict the elements
of each set by interpreting ISETL code. For the union, intersection, and subset
operations, students do much the same: they predict results and then check their
predictions, as shown in Lines 7 through 12. Despite the similarity of the outward
activity, there is a difference in the inward cognitive activity. Writing or
interpreting code requires one to think in terms of a Process, that is, to carry out
in one’s mind the Action of running through the elements of the set to check the set
membership condition. On the other hand, predicting the elements of unions and
intersections, and determining whether one set is a subset of the other, requires one
to think of the sets as mental Objects since forming a union, taking an intersection,
or determining a subset relationship are Actions performed on sets.
Write funcs that Accept Sets as Inputs: Students were asked to define binary
operations on sets. In ISETL, this means writing the code that constructs a functhat accepts two elements of the set and returns the result of the operation. This type
of activity constitutes an Action applied to the set. In order to carry out this Action
successfully, the set would need to be encapsulated into a mental Object. A sample
activity with accompanying code illustrates this:
Example: Write code that constructs a func that accepts two elements of a set and returns
the result of a binary operation defined on that set applied to the two elements. Then, write
code to determine whether the func defined on G is commutative.
Line 1: > G:¼{1..12};Line 2: > G;Line 3: {1, 2, 4, 3, 12, 11, 10, 9, 6, 5, 8, 7};Line 4: > o:¼func(x,y);Line 5: >> if (x in G and y in G) thenLine 6: >> return (x*y) mod 13;Line 7: >> end;Line 8: >> end;Line 9: >
5.3 Teaching and Learning Groups 69
Line10: > is_commutative:¼func(S,op);Line11: >> return forall x,y in S : x .op y ¼ y .op x;Line12: >> end;Line13: >Line14: > is_commutative(G,o);Line15: true;
In Line 1, the set G is defined. Line 2 is an ISETL call for G, which is returned inlist form in Line 3. Lines 4 through 8 give the code for the func o, which is a
binary operation defined on G. In order for a student to write the func o, he/shemust write code to select two arbitrary elements from the set G and then apply the
operation to those elements. The selection of arbitrary elements constitutes an
Action applied to the set. As a result, the set must first be encapsulated into a
mental Object so that the Action can be applied. Lines 10 through 12 are the ISETL
code for the func is_commutative. This func accepts a set S and binary
operation op defined on S. The func tests whether the binary operation opdefined on S is commutative. The func is called for the set G and operation o.Since o is commutative, the func returns true (Line 15). Since the funcis_commutative is an Action applied to its inputs, determining the code for
is_commutative and then applying it may encourage encapsulation of the set
and binary operation concepts.
5.3.1.2 Binary Operation
For a specific finite set, a specific binary operation is an Action applied to the set.
Writing a func in ISETL may lead to interiorization of that Action. An example of
such an activity follows:
Example: For the set S3 of all permutations on three elements, write the code for a funccomp that accepts two elements from S3 and returns the composition of those elements.
Line 1: > S3:¼{[a,b,c] : a,b,c in {1..3} | #{a,b,c}¼ 3};Line 2: > comp:¼func(p,q);Line 3: >> if (p in S3 and q in S3) thenLine 4: >> return [p(q(i)) : i in [1..3]];Line 5: >> end;Line 6: >> end;
The set S3 is defined in Line 1. Lines 2 through 6 give the ISETL code for the
funccomp, which accepts two elements ofS3 and returns the composition of those
permutations. The act of writing this and other binary operation funcs supports theinteriorization of the binary operation Action as the learner begins to think in terms
of general steps: the selection of arbitrary elements together with a means of
assigning another element of the set to the given pair. This type of activity involves
the essence of Process—to move from consideration of specific pairs to thinking
about arbitrary pairs and to think about how a binary operation operates generally.
70 5 The Teaching of Mathematics Using APOS Theory
Interiorization may be supported further by interpretation of quantified
statements. For instance, when asked to reflect on a quantified statement such as
> forall g1, g2 in G | g1 .o g2 in G;
a student needs to think in terms of checking an arbitrary pair. This necessitates
interiorization of the Action of testing the binary operation for a specific pair of
elements in the set G.Testing whether a binary operation satisfies certain properties can also lead to
encapsulation. To determine whether a binary operation satisfies a property, a student
writes a Boolean-valued func that accepts the set and binary operation as inputs, tests
the property for every pair of elements in the set, and returns true or false. Since this
constitutes an Action on both the set and the binary operation, both need to be
encapsulated as Objects. Since the actual test involves a coordination of the set and
binary operation Processes, the two Objects are de-encapsulated so that the property
can be checked for every pair of elements. The example and sample code that follows
illustrates how this might work for the set S3 and the binary operation comp.
Example: Write a func that accepts as input a set and a binary operation defined on that
set, and that tests whether the binary operation is commutative.
Sample Code:
Line 1: > S3:¼{[a,b,c] : a,b,c in {1..3} | #{a,b,c}¼ 3};Line 2: > comp:¼func(p,q);Line 3: >> if (p in S3 and q in S3) thenLine 4: >> return [p(q(i)) : i in [1..3]];Line 5: >> end;Line 6: >> end;Line 9: >Line 10:> is_commutative:¼func(S,op);Line 11: >> return forall x,y in S : x .op y ¼ y .op x;Line 12: >> end;Line 13: >Line 14: > is_commutative(G,o);Line 15: false;
The func is_commutative (lines 10–12) is a Boolean-valued function that
accepts a set S and a binary operation op and checks whether the operation opdefined on S is commutative. In line 14, the func is_commutative is applied
to the set S3 and to the operation comp. The result of that test appears in Line 15.
Several kinds of binary operations appear in ISETL as predefined operations.
These include mod, div, min, max, and, or. Students work with these without
explicit mention that they are binary operations. Activities involving these
commands are designed to help students to construct a binary operation Schema.
In alignment with standard mathematical notation, and as mentioned earlier,
ISETL supports the use of infix notation. If op is any func of two variables
in ISETL, the expression a .op b may be used instead of op(a,b). For the
5.3 Teaching and Learning Groups 71
following example, add_Z20(3,5) (Lines 8–9) returns the same result as
3.add_Z20 5 (Lines 10–11):
Line 1: > Z20:¼{0..19};Line 2: > add_Z20:¼func(x,y);Line 3: >> if (x in Z20 and y in Z20) thenLine 4: >> return (x+y) mod 20;Line 5: >> end;Line 6: >> end;Line 7: >Line 8: > add_Z20(3,5);Line 9: 8;Line 10: > 3 .add_Z20 5;Line 11: 8;
The use of infix notation helps students to see that any function of two variables
defined on a set is a binary operation. This is another feature of ISETL that
promotes development of a binary operation Schema.
5.3.1.3 Group Schema
A group Schema is constructed mentally through coordination of the axiom Schema
with the set and binary operations Schemas. The axiom Schema includes two
principal components:
1. Checking a property of a binary operation defined on a set
2. The four axioms of the group concept constructed as Objects
The general Process of satisfying a property is coordinated with the specific
Process for the axiom, which is de-encapsulated from the axiom Object. This
coordinated Process is then applied to a particular set and binary operation.
This involves de-encapsulation of the set and binary operation Objects followed
by coordination of the set, binary operation, and axiom Processes to establish the
validity of an axiom for a given set, binary operation pair. These types of mental
constructions are facilitated by activities in which students write funcs such as
is_closed, is_associative, has_identity, and has_inverses.Each func accepts a set and a binary operation as inputs and returns a Boolean
result. Students test these funcs on specific set and binary operation pairs. For
example, to test associativity (Lines 1–3) of the function composition operation
comp defined on S3 (Line 5), students would enter the code is_associative(S3,comp) (Line 11), where S3 denotes the set and comp (Line 7–9) represents
the composition operation:
Line 1: > is_associative:¼func(G,op);Line 2: >> return forall x,y,z in G | (x .op y) .op z ¼ x .op
(y .op z);Line 3: >> end;Line 4: >
72 5 The Teaching of Mathematics Using APOS Theory
Line 5: > S3:¼{[a,b,c] : a,b,c in {1,2,3} : #{1,2,3}¼ 3};Line 6: >Line 7: > comp:¼func(p,q);Line 8: >> return [p(q(i)) : i in [1..3]];Line 9: >> end;Line 10:>Line 11:> is_associative(S3,comp);Line 12:true;
The four instances of the coordination of set, binary operation, and axiom that
underlie the closure, associativity, existence of identity, and existence of inverses
axioms are then coordinated into a total Process of satisfying the axioms for a
group. This mental construction is advanced by having students write a funccalled is_group. This func accepts a set and a binary operation and returns a
Boolean value. One possibility for is_group is the following:
> is_group:¼func(G,op);>> return is_closed(G,op) and
is_associative(G,op) andhas_identity(G,op) andhas_inverses(G,op);
>> end;
As students apply is_group to different set and binary operation pairs, they
build a collection of examples and non-examples of groups. Consideration of
different examples supports coherence of the group Schema. This includes the
ability to recognize those relationships that are included in the Schema and to
decide, when facing a problem situation, if the characteristics of the problem are
within the scope of the Schema.
Reflection on the components and relations that make up a Schema enables the
individual to perform conscious Actions on it. The ability to construct such Actions
is an indicator of thematization of the group Schema. Activities and exercises that
promote thematization include determination of whether a particular set and binary
operation forms a group, checking various properties of a group, or considering
whether two groups are isomorphic. As one example of checking properties,
students construct a func is_commutative that accepts a group and its
accompanying binary operation, checks the commutative condition for each pair
of elements, and returns true or false. In a later chapter in the book, students areasked to construct homomorphisms. Central to this construction, which supports
thematization of the group Schema, is construction of a Boolean-valued funccalled is_hom that accepts a representation f of a function between two groups
(G, op) and (G’, op’)and determines whether the homomorphism condition is
satisfied. The is_hom func might take the following form:
> is_hom:¼func(f);>> return forall x,y in G | f(x .op y) ¼ f(x) .op’
f(y);>> end;
5.3 Teaching and Learning Groups 73
5.3.2 Class Discussion
The class discussion phase of the ACE Cycle consists principally of two parts:
full-class consideration of examples and non-examples of groups that were
constructed in the Activities phase and construction of different symmetry groups
that can be represented with geometric models.
In the activities, students construct the following set and binary operation pairs
and then test whether those pairs satisfy certain properties and form groups:
• (Z12, a12), where Z12 is the set of integers {0,1,2,3,4,5,6,7,8,9,10,11} and a12
is addition modulo 12
• (Z12, m12), where Z12 is the set of integers {0,1,2,3,4,5,6,7,8,9,10,11} and m12
is multiplication modulo 12
• (twoZ12, m12), where twoZ12 is the set of integers {0,2,4,6,8,10} and m12 is
multiplication modulo 12
• (Z12-{0}, m12), where Z12-{0} is the set of integers {1,2,3,4,5,6,7,8,9,10,11}
and m12 is multiplication modulo 12
• (S3, op), where S3 is the set of permutations of the set {1,2,3} and op is
composition of permutations
Individual groups of students share their results, with the goal of reaching a class
consensus. Included in these discussions are consideration of the properties of
binary operations and groups.
The class discussion calls for students to construct the group D4, the set of all
symmetries of the square. Students take a square, number its corners, determine the
possible rotations and reflections, form an operation table, verify that the set and
binary operation pair forms a group, and consider the relation between this group
and the set of all permutations of the set {1,2,3,4}. This exercise, which involves
work with a geometric representation, helps the students to expand their under-
standing of the group Schema.
5.3.3 Exercises
The exercises phase of the ACE Cycle reinforces the Activity and Classroom
Discussion phases. Specifically, students continue to build and to expand their
group Schemas by working with sets of matrices, permutation groups, and the
rational numbers and by proving certain properties of groups, particularly
conditions that guarantee that a group is Abelian, as well as showing that every
group of order four is Abelian. The exercises also point to future work with orbits
and subgroups. In one example related to the former, students determine the value
of n for which pn, p 2 S6 (the set of all permutations on six elements), yields
the identity. For the latter, students determine all two, three, and four element
subgroups of S3, the set of all permutations of {1,2,3}.
74 5 The Teaching of Mathematics Using APOS Theory
5.3.4 Results of the Instruction
Brown et al. (1997) reported on the results of students’ learning of groups in courses
based on the APOS instructional approach detailed in Dubinsky and Leron (1994).
With respect to the genetic decomposition, the data seemed to support the
preliminary genetic decomposition. The data also revealed specific issues that
arise in the development of the construction of the coordination between
the axiom Schema and the set and binary operation Schemas. The issues have the
potential to highlight specific aspects of learning about groups that might be helpful
to instructors:
1. At the early stages of coordinating these Schemas, students have a tendency to
assume that a feature that appears in one part of an environment applies
throughout the entire environment (e.g., a student might express the opinion
that a given subset of a group is closed because the group itself is).
2. In working with sets and conditions for set membership, students find it easier to
see that an element satisfies the condition of being a member of the set than it is
for them to grasp the idea that being an element of the set implies that the
condition is satisfied.
3. In their construction of a group Schema, students must construct an understand-
ing of the notion of a generic group and be able to perform calculations therein.
Brown et al. (1997) also conducted a comparative analysis of students who
completed the APOS-based course with students who completed a traditional
abstract algebra course. The students in the APOS group performed better on
mathematical tasks related to binary operations, groups, and subgroups than did
their traditionally instructed counterparts. In addition, the data also showed that at
least one-third of the students who received the APOS instruction succeeded in
understanding the concepts, while most of the rest made significant progress, and all
showed an ability to deal with the material.
5.4 Application of the ACE Teaching Cycle in a Unit
on Repeating Decimals
Rational numbers are studied extensively at the elementary and middle school levels.
As a result, it would stand to reason that elementary and middle school teachers need
to have a strong foundational understanding of rational number concepts, particularly
their representations, which include repeating decimal expansions. However,
Yopp et al. (2011) found that many preservice elementary and middle school teachers
have considerable difficulty with repeating decimals. The participants in their study
asserted the existence of infinitesimals (“there’s a wee bit missing”), that real numbers
correspond directly and solely to physical experiences, and that approximations are
sufficient. Tall and Schwarzenberger (1978) also found that college students think in
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 75
terms of infinitesimals and are confused over the fact that two different decimals
can correspond to the same rational number. Given teachers’ difficulties, it is not
surprising that K–12 students hold misconceptions. Gardiner (1985) reported that
K–12 students often view infinite decimals as being like finite decimals and claim
that the smallest positive number is 0.0. . .01.Repeating decimals (discussed earlier in Sect. 4.2) are one of several mathemat-
ical conceptions that stand at the nexus of the paradoxical duality between potential
and actual infinity. On one hand, a repeating decimal can be thought of as an
instance of potential infinity—a process of continually forming digits to express a
rational number through long division. On the other hand, a repeating decimal is an
instance of actual infinity—the representation of a number with fixed value.
Dubinsky et al. (2005a, b) studied the apparent tension between these seemingly
contradictory notions in an APOS-based analysis of the historical development of
the concept of mathematical infinity. In their analysis, they explained how potential
and actual infinity represent two different conceptualizations linked by the mental
mechanism of encapsulation. Potential infinity, the notion of infinity presented over
time, is the conception of infinity as a Process. Because an infinite process has no
final step, and hence no obvious indication of completion, the ability to think of an
infinite process as mentally complete is a crucial step in moving beyond a purely
potential view. As an individual reflects on a completed infinite process, he or she
can conceive of it as a totality, a single operation freed from temporal constraints.4
At this point, the individual can apply the mechanism of encapsulation to transform
the Process into a mental Object, an instance of actual infinity.
The authors used these ideas to explain individuals’ difficulties with the repeat-
ing decimal 0:�9 and 1. An individual may think of 0:�9 as a Process, a dynamic view
of continually adding 9s, or something one does, whereas he or she may conceive of
the number 1 as a mental Object, a static entity that can be transformed. Given that
it does not make sense, at least mentally speaking, to compare a dynamic Process
with a static Object, one who sees 0:�9 as a Process and 1 as a mental Object may see
0:�9 and 1 as being unequal. Alternatively, an individual may view an infinite
repeating decimal as an incomplete Process. In this case, a repeating decimal is a
finite string of digits with indeterminate length. With such a conception, one might
think of 0:�9 as infinitesimally close to but less than 1.
This analysis was used to develop a preliminary genetic decomposition for
infinite repeating decimals that informed the design of an instructional unit on
repeating decimals for preservice elementary and middle school teachers. A student
begins by constructing certain Actions on whole numbers. This involves reciting,
either verbally or in writing, an initial sequence of digits, which may be seen as the
beginning of a repeating decimal expansion. These Actions are interiorized into a
4 In Dubinsky et al. (2005a, b), the ability to see a Process as a Totality was considered to be a part
of encapsulation. The instruction on which the study was based (Weller et al. 2009, 2011;
Dubinsky et al. 2013) showed evidence of Totality as a separate stage between Process and Object.
This distinction is considered later in this chapter and explored in depth in Chap. 8.
76 5 The Teaching of Mathematics Using APOS Theory
Process of forming sequences of digits of indeterminate length that is extended to
form an infinite string. Specifically, the student grasps the idea that from some point
on the decimal repeats forever to form an infinite string. As the student reflects on the
Process and begins to see an infinite string as an entity to which mental Actions or
Processes can be applied, the Process of forming an infinite stringmay be encapsulated
into a mental Object. The Actions that may be applied to an infinite string
include various arithmetic and comparison operations, determination of whether an
infinite string satisfies certain relations or arithmetic equations, and the ability to see
a repeating decimal as a number that equals a fraction or integer. The preliminary
genetic decomposition for repeating decimals is illustrated in Fig. 5.4.
The instructional treatment of repeating decimals consisted of three iterations of
the ACE cycle. Each iteration of the cycle spanned two class days, one for computer
activities and one for classroom discussions. Homework exercises were assigned at
the end of each session and collected at the beginning of the next session. Two
additional classroom meetings were reserved for time overruns. Analysis of this
instructional unit appears in three reports, Weller et al. (2009, 2011) and Dubinsky
et al. (2013).
For the activities phases of the cycle, students worked in cooperative groups in a
computer lab where they used ISETL. In this particular instance, the use of ISETL
differed from previous instruction based on APOS Theory. Typically, students use
ISETL to write short computer programs. This type of activity supports the mental
mechanisms that lead to construction of the mental structures called for by a genetic
decomposition. For instance, writing and then reflecting on programs that carry
out Actions supports interiorization. Writing programs that perform Actions on
Processes supports encapsulation. For the instructional sequence on repeating
decimals, the students performed calculations in ISETL using a preloaded decimal
expansion package developed by the researchers (Weller et al. 2009, 2011;
Dubinsky et al. 2013). Although the students were not asked to write computer
programs, the calculations supported the mental mechanisms of interiorization and
encapsulation. Students used preloaded funcs to look at a single place or finite
range of places of a repeating decimal. This type of activity supported interioriza-
tion by helping students to reflect on the Action of writing out the terms of a
decimal expansion. The students used predefined funcs to perform arithmetic
operations and comparisons on repeating decimals and fraction-to-decimal and
decimal-to-fraction conversions. These types of activities supported encapsulation
Fig. 5.4 Diagram of the genetic decomposition of infinite repeating decimals
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 77
by having students perform Actions on repeating decimals. In addition to
predefined funcs, the decimal expansion package stored several examples of
repeating decimals for use in different activities. For many of the activities, students
performed calculations by hand and then checked their results with the computer.
Examples of these types of activities are given in Sects. 5.4.1–5.4.3.
The purpose of the first iteration of the ACE Cycle was twofold: to help students
(1) to interiorize the Action of listing digits to a mental Process (in order to conceive
of an infinite string5 of digits comprising a repeating decimal) and (2) to begin to
see a repeating decimal as a mental Object by agreeing on a notational scheme for
its representation. The second iteration of the cycle focused on encapsulation—to
help students to transform infinite digit strings conceived as Processes into mental
Objects to which Actions could be applied. The third iteration emphasized devel-
opment of the relation between an infinite digit string and its corresponding fraction
or integer. Development of this relation is an important part of an individual’s
rational number Schema. The construction and subsequent encapsulation of differ-
ent rational number representations enables an individual to expand her or his
Schema, offers the potential to develop the coherence of the Schema, and increases
the likelihood for an individual to see a rational number as an entity that has value
that can be compared with other numbers, that has a position on the number
line, etc.
The instructional treatment did not rely on limits. Although preservice elemen-
tary and middle school teachers who specialize in the teaching of mathematics
typically complete a calculus course, the instructional treatment was designed for
a content course on number and operation required by all preservice elementary
and middle school teachers, regardless of their area of specialization. Given
that numerous studies have documented college students’ difficulties with limits
(e.g., Cornu 1991; Cottrill et al. 1996; Sierpinska 1987; Williams 1991, 2001),
the APOS-based instructional treatment avoided calculus treatment of limits as well
as instruction on conversion techniques that assume knowledge of limits and
infinite series. Each of the three iterations of the ACE Cycle is described in detail
in the sections that follow.
5.4.1 First Iteration of the Cycle
The first iteration of the ACE Cycle was designed to encourage the development
of a Process conception of a repeating decimal and to begin to help students to see
a repeating decimal as a mental Object by considering notational schemes to
represent them.
5 In these discussions, decimal expansions are referred to as strings. This means finite or infinite
sequences of digits that correspond to the decimal expansion of a rational number.
78 5 The Teaching of Mathematics Using APOS Theory
5.4.1.1 Activities
In the computer lab, students completed activities using eight preloaded decimal
expansions6 whose identity7 was purposely left a mystery. In the activities with
these strings, students made extensive use of a predefined func called View. TheView command works in two ways: View(ds,n) yields the nth place of the
decimal string ds and View(ds,n,k) returns the nth through kth entries of ds.The underlying idea behind the func View was its potential to support interiori-
zation of the Action of listing the digits of a decimal string. This goal may be more
apparent by considering a specific example. To determine the identity of the
mystery string m5, one might apply View to several individual values of n:
> View(m5,1);"1";> View(m5,2);"0";> View(m5,3);"3";> View(m5,4);"5";> View(m5,5);"8";> View(m5,6);"5";> View(m5,7);"8";> View(m5,-1);"1";> View(m5,-2);"2";> View(m5,-3);"";
For positive values of n, View returns the nth place after the decimal point. For
negative values of n, View returns positions to the left of the decimal point: n ¼ �1
corresponds to the one’s place, n ¼ �2 to the ten’s place, etc. The fact that Viewreturns 5 for n ¼ 4, 8 for n ¼ 5, 5 for n ¼ 6, and 8 for n ¼ 7 suggests the possibility
of a repeating decimal involving the digits 5 and 8 that starts in the ten-thousandth’s
6 ISETL recognized decimal expansions using the notation a.b(c). Here a, b, and c are nonnegativeintegers, where a denotes the integer part of the decimal expansion, b the decimal portion that
appears before the repeating cycle, and c the repeating cycle. For repeating digits such as 0:�3
and 0:35, where the cycle begins in the tenths place, the computer recognized the notation 0.3(3)
and 0.3(53), respectively.7 The preloaded mystery strings were denoted m1, m2, m3, m4, m5, m6, m7, and m8.
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 79
place. The fact that View returns “ ” for n ¼ �3 and n ¼ �4 suggests that all place
values to the left of the ten’s place are zero. Based on the entries, it appears that
m5 might be the decimal 21.1035858. . .. To obtain a sense of whether this is the
case, one might consider View for a range of values, say for n ¼ �3 to k ¼ 20 and
then for n ¼ �3 and k ¼ 40:
> View(m5,-4,20);"21.10358585858585858585";> View(m5,-5,40);"21.1035858585858585858585858585858585858585";
Beyond merely substituting different values for n and k, identification activities
included questions such as the following:
What are the digits in the first 10 places to the right of “.”?
What digit appears in the 100th place after the “.”?
What digits appear in places 101, 102, 103, 104, 105 after “.”?
What digit appears in position 1034 after “.”? What are the digits in neighboring places?
What digits appear in places 100000, 100001, 100002, 100003, 1000004, 100005 after “.”?
Using the information you have gathered, write a formula to determine the identity of the
digit in the nth position of the decimal string.
This type of exploration and consideration of these types of questions were
central to the goal of leading the student from external representation, the act of
listing digits, to an internal image, the ability to imagine a digit in every position to
the right of the decimal point.
Following questions on identification, students responded to more general
questions regarding View as a means of encouraging deeper reflection:
What do you get from View if n ¼ 0?
What is the significance of what View returns if n is positive? Negative?
What is the significance of what View returns if n is very large and positive? Very large
and negative?
In addition to returning any position in a decimal expansion, the func Viewhelps students to think of a decimal as an infinite string that extends indefinitely
in either direction from the decimal point. The decimal point is returned by Viewwhen n ¼ 0. Thus, for n < 0, View returns integer place value positions, and, for
n > 0, View returns decimal positions. The following example for the mystery
string m3, with n ¼ �5 and k ¼ 20, helps to exemplify this8:
> View(m3,-5,20);"11.14285714285714285714";The activities involving prediction and reflection culminated with two overarching
questions: (1) What is a repeating decimal? (2) Which of the preloaded strings would
you consider to be repeating decimals? Again, the idea was to encourage interioriza-
tion by helping students to think about a decimal string as an unending list of digits,
which, in the case of repeating decimals, involves a pattern of repetition.
8 Because positions n ¼ �3;�4;�5 are 0, they do not appear, according to convention.
80 5 The Teaching of Mathematics Using APOS Theory
As a means of encouraging encapsulation, the final activity asked the students to
figure out a notational scheme for repeating decimals, with particular focus on
notation that would be used to represent decimal expansions in ISETL.
5.4.1.2 Class Discussion
Since the purpose of the first iteration of the cycle was to facilitate the development
of a Process conception of decimal strings, the instructor’s goal was to support
reflective activity that would help students to interiorize the external Action of
listing the digits of a decimal string. Beyond having the students share their
descriptions of the mystery strings, this called for the class discussion to focus
squarely on consideration of the functioning of View, with particular attention paidto what View returns for different values of n. The class discussion also included
introduction of the notion of the meaning of decimal string. This discussion
culminated with consideration of the question of “What is a repeating decimal?”
To begin to move students toward encapsulation, the instructor engaged the class
in a discussion regarding notation. The instructor invited student groups to share
their notational schemes leading to a class notational scheme. This discussion
served as the basis for consideration of what ISETL accepts.
5.4.1.3 Exercises
Several of the exercises involved application of the command View to determine
unknown strings. Students were presented with several lines of output and asked to
provide a description expressed in the notation agreeduponduring the class discussion.
Additional exercises asked for descriptions of unknown strings subject to certain
conditions. Although only partial information was provided, it was sufficient to
specify a string uniquely. The example below captures the essence of this type of
exercise:
Exercise: Describe the decimal string str3 if the following conditions must be satisfied:
a. The digits 223 go before the decimal point.
b. The first string position after the decimal point is equal to 5.
c. The second string position after the decimal point is equal to 0.
d. The third string position after the decimal point is equal to 4.
e. The fourth string position after the decimal point is equal to 6.
f. For every integer n � 0, the digit in the 2nþ 5 position after the decimal point is equal to 8.
g. For every integer n � 0, the digit in the 2nþ 6 position after the decimal point is equal to 3.
In addition to your description, express str3 using the notation devised in class.
The final exercise asked students to offer their own explanation of how Viewworks. Although discussed in class, this exercise provided an opportunity for
students to reflect on the construction of a decimal string as they described the
ISETL command.
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 81
5.4.2 Second Iteration of the Cycle
The second iteration of the cycle was designed to help students make
encapsulations—to transform infinite digit strings conceived as Processes into
mental Objects to which Actions could be applied. Thus, the work in this session
focused on having students perform a variety of operations on strings.
5.4.2.1 Activities
In the computer lab, the students were presented with 16 infinite strings (whose
identity was revealed, unlike the first iteration) that were represented as repeating
decimals and stored in the computer in the form a:bðcÞ. In the first set of activities,
students performed standard arithmetic operations by hand on different pairs of
decimals from the predefined list. They then used ISETL to compare the results of
their written work. Students used commands from the decimal expansion package
developed by the researchers to carry out their operations. Examples of the
16 predefined strings, along with operations performed on them, appear below:
Line 1: > ds1;Line 2: "0.23(4)";Line 3: > ds2;Line 4: "2.125(0)";Line 5: > ds3;Line 6: "11.1(428571)";Line 7: > ds4;Line 8: "0.1(41)";Line 9: > ds5;Line 10: "0.7(867)";Line 11: > ds6;Line 12: "10.0(0)";Line 13: > ds7;Line 14: "100.0(0)";Line 15: >Line 16: > ds1 .AddString ds2;Line 17: "2.359(4)";Line 18: >Line 19: > ds1 .SubString ds4;Line 20: "0.09(30)";Line 21: >Line 22: > ds5 .MultString ds6;Line 23: "7.8(678)";Line 24: >Line 25: > ds3 .DivString ds7;Line 26: "0.11(142857)";
82 5 The Teaching of Mathematics Using APOS Theory
The variables ds1, ds2, ds3, ds4, ds5, ds6, and ds7 (Lines 1–14) denote
some of the decimal strings. The remaining lines of code show operations on those
strings: AddString (Lines 16–17) represents addition, SubString (Lines
19–20) denotes subtraction, MultString (Lines 22–23) stands for multiplication,
and DivString (Lines 25–26) signifies division. Each of the string operations are
funcs that are part of the decimal expansion package.
The next set of activities called for comparisons. Students ordered the
16 predefined strings by hand, checked their work using comparison commands
included in the decimal expansion package, and explained how one determines
whether one string is larger than another. Another related activity called for
students to make comparisons after applying arithmetic operations. For example,
given the three decimal strings s1¼0.1(41), s2¼0.5(0), and s3¼0.2(132),students determined by hand whether the sum of s1 and s3 exceeds the sum of
s1 and s2. Once they performed these computations, they tested their results
using ISETL:
> s1:¼"0.1(41)";> s2:¼"0.5(0)";> s3:¼"0.2(132)";>> (s1 .AddString s2) .LessString (s1 .AddString s3);false;
The activity involved carrying out arithmetic operations (via AddString),order of operations (through use of parentheses), and comparison (by application
of LessString). LessString, as the name suggests, is a Boolean operator that
tests whether the first entry (in this case, s1 .AddString s2) is smaller than the
second (s1 .AddString s3). It was one of three comparison funcs defined by
the researchers and included in the decimal expansion package. The other two
funcs, EqualString, which tests whether two decimal strings are equal, and
GreatString, which tests whether the first entry is larger than the second, are
also Boolean operators.
The remaining activities emphasized the connection between representations,
that is, that each fraction or integer has a corresponding decimal expansion.
The researcher-developed decimal expansion package included a func called
Dec2Frac that accepts a string and returns the fraction that corresponds to the
given string. The command Frac2Dec does the reverse—given a fraction, it
returns the corresponding decimal expansion. For the given list of 16 strings,
students were asked to think of a number (fraction or integer) that would correspond
to each string. They then applied Dec2Frac to each string in the list to test their
predictions. The students were then asked to perform long division on the fractions
returned by ISETL to verify by hand the correspondence revealed by Dec2Frac.In the code given below, the func Dec2Frac shows that the repeating decimal
0:23�4 corresponds to the fraction 211900
and that the decimal 0:7867 corresponds to the
fraction 262333
.
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 83
> Dec2Frac("0.23(4)");[211, 900];>> Dec2Frac("0.7(867)");[262, 333];
This correspondence/conversion activity set the stage for two questions: (1)
What are the characteristics of a number, that is, what makes a number? (2) In
what sense do decimal strings fit your description of a number? Both of these
questions encouraged the students to reflect on the operations they had performed,
to begin to see repeating decimals as Objects that correspond to other Objects
(fractions and integers), and to which operations can be applied.
5.4.2.2 Class Discussion
Since the purpose of the second iteration of the cycle was to facilitate encapsula-
tion, the instructor’s role was to help students see how decimal strings behave like
numbers, and to consider why it is reasonable to attach integers or fractions to
decimal strings. The former involved reflection on the arithmetic operations the
students performed in their lab activities, specifically, a discussion regarding
the characteristics of number and the way in which the activities demonstrated
how decimal strings behave like numbers. The latter required a discussion of
the conversion funcs, Dec2Frac and Frac2Dec, with the goal of helping the
students to see the correspondence between decimal strings and integers and
fractions.
The activities and supporting class discussion also opened the door for consid-
eration of the connection between the repeating decimal 0:�9 and 1, which was dealtwith at great length in the third and final iteration of the ACE Cycle.
5.4.2.3 Exercises
In the exercises, students performed arithmetic operations using both repeating
decimal and fraction/integer representations as a means of seeing the connection
between the different representations. Specifically, they solved by hand simple
equations of the form s :op t ¼ x, where s and t are strings (in the form a:bðcÞ)and op is an arithmetic operation. Then they performed the same arithmetic
operations using the integer/fraction representations for s and t. After making
these computations, they compared the representations of the result x by long
division (where they transformed x from integer/fraction to decimal form) and
through use of the ISETL func Dec2Frac (to convert the string representation
of x to its fraction/integer form) to see that the operations yield equivalent results
no matter the representation. Students were also given a list, in random order, of six
strings and a second list, in a different random order, of six fractions. They ordered
84 5 The Teaching of Mathematics Using APOS Theory
both lists by size and then determined the correspondence between the individual
strings and the fractions, first by comparing the order and then by long division.
These exercises were designed to encourage encapsulation: by having students
connect repeating decimal expansions with familiar representations that were
more likely to be seen as Objects and by reinforcing the idea that repeating
decimals, when operated on, behave like other numbers.
5.4.3 Third Iteration of the Cycle
The third iteration of the cycle emphasized the relation between infinite decimal
strings and their corresponding fraction or integer representations, with special
attention paid to the relationship between 0:�9 and 1. This focus supported encapsu-
lation in three ways: (1) by connecting infinite strings with fraction or integer
representations that are more likely to be seen as Objects; (2) by performing
Actions on infinite decimals, in which strings are converted to their fraction or
integer equivalents; and (3) by seeing that infinite decimal strings yield the same
results in arithmetic operations as their corresponding fraction/integer counterparts.
5.4.3.1 Activities
In the computer lab, the students were asked to perform long division on fractions
and to use commands from the ISETL decimal expansion package to check their
answers. In one such activity, students were asked to perform long division on the
fractions 27; 4
9; 3
11; 2
13and then apply the command Frac2Dec to verify their
findings. After making these computations, they explained why each string,
obtained either by hand or by application of the ISETL command, is equal to the
fraction on which they performed long division. The ISETL code for making these
conversions follows:
> Frac2Dec(2,7);"0.2(857142)";>> Frac2Dec(4,9);"0.4(4)";>> Frac2Dec(3,11);"0.2(72)";>> Frac2Dec(2,13);"0.1(538461)";
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 85
These lines of code verify the results of the long division carried out by hand.
The reverse process, moving from decimal string to fraction, is illustrated next:
> Dec2Frac("0.2(857142)");[2, 7];>> Dec2Frac("0.4(4)");[4, 9];>> Dec2Frac("0.2(72)");[3, 11];>> Dec2Frac("0.1(538461)");[2, 13];
In a second series of activities, students performed a number of arithmetic and
ordering operations, first on strings and then on the corresponding fraction/integer
representations. They completed this work by hand and then used commands from
the ISETL decimal expansion package to check their answers. In each case, they
reflected on whether the fraction/integer representations are equal to the
corresponding strings. For example, for sums, students were given the decimals
0.3125 and 0:�1 and their corresponding fraction representations 516
and 19. They
applied Dec2Frac to each decimal and Frac2Dec to each fraction to see the
connection between the different representations. Then, they found the sums, first
for the two decimals and then for the two fractions. After making these calculations,
they applied Frac2Dec and Dec2Frac to their results to see that the operations
yield equivalent results regardless of the representation being worked with. The
ISETL code is shown below:
> Dec2Frac("0.3125(0)");[5, 16];>> Dec2Frac("0.1(1)");[1, 9];>> "0.3125(0)" .AddString "0.1(1)";"0.4236(1)";>> 5/16 + 1/9;61/144;>> Dec2Frac("0.4236(1)");[61, 144];>> Frac2Dec(61,144);"0.4236(1)";
86 5 The Teaching of Mathematics Using APOS Theory
These lines of codes are reflected in a commutative progression illustrated in
Fig. 5.5.
Figure 5.5 shows a generalization of the steps of the procedure given in the ISETL
example. Specifically, for two fractions ab and c
d and their corresponding decimal
expansions, the ISETL commands Frac2Dec and Dec2Frac transform one
representation into another, that is, Frac2Dec returns the decimal expansion that
corresponds to a given fraction andDec2Frac returns the fraction corresponding to
a given decimal. The left side of the diagram shows the addition of the two fractional
representations. The right side of the diagram shows the sum of the decimal
expansions using the ISETL command .AddString. The bottom of the diagram
shows how the ISETL commands Dec2Frac and Frac2Dec can be used to show
the correspondence between the sums of the two representations.
Carrying out decimal-to-fraction and fraction-to-decimal conversions, both by
hand and with ISETL, and showing the equivalence of the results of arithmetic
operations, no matter the representation, was seen as a means of supporting
encapsulation of repeating infinite decimal representations.
The next series of activities dealt with repeating nines. In one of these activities,
the students were given several lines of code in which a terminating decimal with 9s
is subtracted from the decimal equivalent of a familiar fraction. The subtraction was
carried out with the func SubString, which was part of the decimal expansion
package. The purpose of the activity was to help the students to see that a fraction
with a terminating decimal expansion could also be represented by a repeating
decimal with an infinite sequence of 9s. The following activity exemplifies this.
Activity: Suppose the following appears on an ISETL screen:
> Frac2Dec(1,2) .SubString "0.4(0)";"0.1(0)"> Frac2Dec(1,2) .SubString "0.49(0)";"0.01(0)"> Frac2Dec(1,2) .SubString "0.499(0)";"0.001(0)"
Fig. 5.5 Verification of preservation of operations for different representations
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 87
> Frac2Dec(1,2) .SubString "0.4999(0)";"0.0001(0)"
1. Use ISETL to continue this for several steps.
2. Use your results together with the screen output that appears above, to answer the
following questions:
a. Do you agree with the statement that each of 0.4, 0.49, 0.499, 0.4999 is close to 12?
Do the decimal strings get closer to 12as you increase the number of decimal places?
b. If we imagine taking the entire decimal string, what is the relation between 12and
0.499999999. . .?
Building on the exercises from the second iteration of the cycle, students completed
activitieswhere theyperformedoperations onequivalent representations. For instance,
they solved for x equations of the form axþ b ¼ cwhere a, b, and cwere first given infraction/integer form and then in decimal form. This enabled comparisons in which
students could see that equivalent representations yield equivalent solutions. The
special case of 0:�9 was included. This is illustrated in the following activity that
involves use of the predefined funcs MultString and DivString that are used
to carry out multiplication and division operations, respectively, on decimal strings.
Activity: Perform the following calculations using the ISETL code .MultString:
0:0 769230ð Þ � 0:�9
(type “0.0(769230)” .MultString “0.9(9)”; and press enter)
19
15� 0:�9
(type Frac2Dec(19,15) .MultString “0.9(9)”; and press enter)
253� 0:�9
(type Frac2Dec(253,1) .MultString “0.9(9)”; and press enter)
0:2ð342Þ � 0:�9
(type “0.2(342)” .DivString “0.9(9)”; and press enter)
713� 0:�9
(type Frac2Dec(713,1) .DivString “0.9(9)”; and press enter)
It would be very cumbersome to try to perform the operations above by hand. However,
it would be very easy to do them by hand if you replaced “0.9(9)” by “1.0(0)”. Explain why
or why not you think this would be correct.
MultString and DivString, two funcs included in the decimal expansion
package, accept two decimal strings and return the product and quotient, respec-
tively. Sample lines of code appear below:
> "0.0(769230)" .MultString "0.9(9)";"0.0(769230)";>> "0.2(342)" .DivString "0.9(9)";"0.2(342)";
88 5 The Teaching of Mathematics Using APOS Theory
Activities such as these, where 0:�9 is substituted for 1 in arithmetic operations,
were developed to support students’ reflection on the relationship between 0:�9 and 1.
5.4.3.2 Class Discussion
To promote further the goal of students’ understanding of the relation between an
infinite decimal string and its corresponding fraction or integer, the class discussion
focused on five “reasons to believe”:
1. Answers in ISETL: For any pair of decimal and fraction representations that
stand for the same rational number value, the funcs Dec2Frac and
Frac2Dec illustrate the correspondence.
2. Performing long division on fractions: Performing long division is an action on a
fraction that gives the corresponding decimal as a result.
3. Approximation involving initial segments of a decimal representation: Succes-
sively smaller arithmetic differences between a rational number representation
and the initial segments of a repeating decimal suggest equality between the two.
4. Effect on operations: A fraction and its corresponding decimal string have the
same effect on the results of various arithmetic operations.
5. Solutions to algebraic equations: Solutions for x of equations such as ax ¼ b areequal in value regardless of the representation.
In addition to more general considerations, the instructor led a discussion
regarding the relationship between 0:�9 and 1. Students were invited to express
their belief regarding the equality and to offer reasons for that belief. This provided
a means by which students on each side of the argument could try to convince their
peers. The goal of this discussion was to generate a list of justification statements to
mirror the more general discussion that preceded it.
5.4.3.3 Exercises
In the exercises, students compared operations on strings with those performed on
their corresponding fraction/integer representations. They approximated fractions/
integers using sequences of finite strings and considered what happens when one
passes to the entire string. The exercises also included items where students
converted a repeating decimal to a fraction or integer and vice versa. They first
performed these computations by hand and then verified their results using the
ISETL decimal expansion package.
Students were also asked to reflect on the special case of the relationship
between 0:�9 and 1. In one of the exercises, students had to determine whether
there is a decimal string between 0:�9 and 1 and, if not, to consider what this suggestsregarding equality. In another exercise, students wrestled with the fact that one
cannot obtain 0:�9 from 1 via long division, although other Actions suggest equality.
In encouraging reflection on this dilemma, students wrote a short essay in which
they were asked to offer a rationale for the equality.
5.4 Application of the ACE Teaching Cycle in a Unit on Repeating Decimals 89
5.4.4 Results of the Instruction
Two studies (Weller et al. 2009, 2011) reported on results of a comparative analysis
of the APOS-based instructional sequence with traditional instruction on repeating
decimals. A third study (Dubinsky et al. 2013) analyzed students’ thinking from the
perspective of the genetic decomposition.
The subjects for the studies included 204 students (77 in APOS-based instruc-
tion; 127 in traditional instruction) who were enrolled at a major university in the
southern United States. The instruction for the repeating decimals unit, which took
place after the final drop date, was part of a required course on number and
operation for preservice elementary and middle school teachers.
In the first study, Weller et al. (2009) compared the gains in procedural and
conceptual understanding of the two groups. They discovered that students who
received the APOS-based instruction made substantial gains when compared with
students who had received the traditional instruction, particularly in their concep-
tual grasp of infinite repeating decimals.
The second comparative study (Weller et al. 2011), conducted several months
after the instructional sequence, focused on the strength and stability of the
students’ beliefs. The analysis revealed that students who received the APOS-
based instruction developed stronger and more stable (over time) beliefs that a
repeating decimal is a number, a repeating decimal has a fraction or integer to
which it corresponds, a repeating decimal equals its corresponding fraction or
integer, and 0:�9 ¼ 1.
The third study (Dubinsky et al. 2013) sought to answer two questions:
1. Does the genetic decomposition provide a relatively objective and reasonable
explanation of student thinking about 0:�9, or does the data suggest revision of thegenetic decomposition?
2. How does progress in the genetic decomposition relate to belief in the equality
0:�9 ¼ 1?
The analysis showed the need to revise the genetic decomposition, specifically,
it called for the introduction of a new stage, Totality (discussed in Chap. 8), as an
intermediate stage between Process and Object. The data also suggested the need
for a finer-grained decomposition to understand the progression from Action to
Process, from Process to Totality, and from Totality to Object. This led to the
introduction of levels (also discussed in Chap. 8) to describe the transition from one
stage to the next. The revised genetic decomposition described the development
from Action toward Object for 83% of the students. Therefore, Dubinsky et al.
(2013) supported a positive response to the first research question (Item 1 above)
when considered in the context of the refined genetic decomposition.
With regard to the second research question (Item 2), the data revealed that the
participants expressed belief or disbelief in ways that correlated with their emerging
conceptions of 0:�9. This showed, at least for the data for this study, that students
90 5 The Teaching of Mathematics Using APOS Theory
who gave evidence of Totality or Object tended to believe more readily in the
equality.
Several authors (e.g., Yopp et al. 2011) argue that preservice teachers need to
understand rational number concepts to avoid teaching false notions to their
students. The APOS-based instructional unit on repeating decimals made progress
in this regard, particularly when compared with traditional instruction on the topic.
5.5 Analysis of Instruction Using the Research Framework
Each implementation of an instructional sequence provides an opportunity to gather
data. The analysis of the data has two purposes: to gauge students’ mathematical
performance, that is, how much mathematics the students learned as a result of the
instruction, and to determine whether the students made the mental constructions
called for by the preliminary genetic decomposition. This provides an opportunity
for researchers to test empirically the preliminary genetic decomposition and to
evaluate the effectiveness of the APOS-based instruction, particularly when com-
pared with other instructional approaches. Issues involving the framework, that is,
research involving the analysis of data, are the subject of Chap. 6.
5.5 Analysis of Instruction Using the Research Framework 91
Chapter 6
The APOS Paradigm for Research
and Curriculum Development
In the Merriam-Webster online dictionary, the word paradigm is defined in the
following way: “a philosophical and theoretical framework of a scientific school
or discipline within which theories, laws, and generalizations and experiments
performed in support of them are formulated; broadly: a philosophical or theoreti-cal framework of any kind.” This definition reflects the contemporary meaning of
the term coined by Kuhn (1962), who spoke of two characteristics of a “paradigm”:
A theory powerful enough to “attract an enduring group of researchers” (p. 10) and
to provide enough open ends to sustain the researchers with topics requiring further
study. In light of these considerations, the overarching research stance linked to
APOS Theory is referred to as a paradigm, since (1) it differs from most mathemat-
ics education research in its theoretical approach, methodology, and types of results
offered; (2) it contains theoretical, methodological, and pedagogical components
that are closely linked together; (3) it continues to attract researchers who find it
useful to answer questions related to the learning of numerous mathematical
concepts, and (4) it continues to supply open-ended questions to be resolved by
the research community.
Some, but not all, studies that adopt APOS as a theoretical framework make use
of all the elements of the paradigm. Depending on the particular project, the reasons
may be methodological or practical, and it would be impractical to consider all
variations of the way in which the APOS paradigm is used in mathematics education
research. What we are describing as the methodological framework in this chapter
can be considered as an “ideal” organization of an APOS-based research study.
6.1 Research and Curriculum Development Cycle
An APOS-based research and/or curriculum development project involves three
components: theoretical analysis, design and implementation of instruction, and
collection and analysis of data. Figure 6.1 shows how these components are related.
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_6,© Springer Science+Business Media New York 2014
93
According to this paradigm, research starts with a theoretical analysis of the
cognition of the mathematical concept under consideration. This gives rise to a
preliminary genetic decomposition of the concept. As discussed in Chap. 4, a
genetic decomposition is a description of the mental constructions and mental
mechanisms that an individual might make in constructing her or his understanding
of a mathematical concept.
As indicated by the arrows in Fig. 6.1, the three components of the research
cycle influence each other. The theoretical analysis drives the design and imple-
mentation of instruction through activities intended to foster the mental
constructions called for by the analysis. Activities and exercises are designed to
help students construct Actions, interiorize them into Processes, encapsulate
Processes into Objects, and coordinate two or more Processes to construct new
Processes. A variety of pedagogical strategies such as cooperative learning, small
group problem solving, and even some lecturing can be highly effective in
helping students learn the mathematics in question. The implementation of
instruction provides an opportunity for the collection and analysis of data,
which is carried out using the theoretical lens of APOS Theory. The purpose of
the analysis is to answer two questions: (1) Did the students make the mental
constructions called for by the theoretical analysis? (2) How well did the students
learn the mathematical content? If the answer to the first question is negative, then
the instruction is reconsidered and revised. If the answer to the first question is
positive and the answer to the second question is negative, the theoretical analysis
is reconsidered and revised. In either case, the cycle is repeated until these
questions are answered positively and the instructor/researcher is satisfied that
the students have learned the mathematical concepts sufficiently well. In other
words, the cycle continues until the empirical evidence and theoretical analysis
point towards the same mental constructions. Finally, as part of its conclusions,
each study offers pedagogical suggestions for implementation and directions for
future research.
The component related to theoretical analysis is explained in detail in Chaps. 3
and 4, while the component related to design and implementation of instruction is
the topic of Chap. 5. The remaining component, namely, collection and analysis of
data, is discussed in detail in this chapter.
Fig. 6.1 Research cycle (adapted from Asiala et al. 1996)
94 6 The APOS Paradigm for Research and Curriculum Development
6.2 Data Collection and Analysis
The data collection and analysis phase is crucial for APOS-based research, since
without empirical evidence, a genetic decomposition remains merely a hypothesis.
Asmentioned in Sect. 6.1, the purpose of data analysis is to answer two questions: (1)
Did the students appear to make the mental constructions described by the genetic
decomposition? (2) How well did the students learn the concept in question?
Different kinds of instruments are used to investigate these two questions.
Depending on the objectives of the particular study, these may include written
questionnaires, semi-structured interviews (audio- and/or videotaped), exams,
and/or computer games. The methodological design may also include classroom
observations, textbook analyses, and historical/epistemological studies. Examples
from the literature that illustrate the kinds of instruments that form part of the
methodological design of a research study are considered in this chapter.
In all of these cases, the analysis is triangulated through collaborative research,
as researchers negotiate results until they reach consensus on their interpretations
and/or by implementing more than one research instrument for a study.
6.2.1 Interviews
Interviews are the most important means by which data is gathered in APOS-based
research. Although an interview may be used to gauge students’ attitudes and to
compare mathematical performance among different approaches to instruction, the
main objective is to determine whether students have made the mental constr-
uctions set forth by the genetic decomposition used in a particular study.
Interview subjects may be selected on the basis of their responses to a written
questionnaire or a previously administered exam, instructor feedback, or a combi-
nation of these criteria. The idea is to access data that shows a range of mathemati-
cal performance on different mathematical tasks in order to compare the thinking of
students who had difficulty with the thinking of students who succeeded. These
differences enable the researchers to determine whether the mental constructions
called for by the theoretical analysis account for differences in performance or
whether other mental constructions not accounted for by the theoretical analysis
are called for.
In designing the interview questions, different sources may be used. The
responses to a previously administered written exam or questionnaire might form
the basis of such questions. In this case, students are asked to clarify their responses
and/or to expand on them. Another possibility might be pilot interviews whose
results might uncover certain issues that can be probed more deeply through an
interview protocol that is administered to a larger group of students. Observations
may also play a role. In this case, difficulties that arise in lab sessions, in classroom
discussions, or in homework exercises may be used as a basis for the construction of
6.2 Data Collection and Analysis 95
interview questions. In all of these instances, the genetic decomposition informs the
design; the goal of the analysis is to determine whether the students made the
proposed mental constructions and to find supporting evidence.
In APOS-based research, as in most semi-structured interviews, the interviewer
follows a prepared outline of questions. Depending on a student’s responses, the
interviewer might ask follow-up questions. The interviewer asks these types of
questions to seek clarification or to probe a student’s thinking more deeply. If these
questions fail to elicit sufficient responses, the interviewer may take a more
didactical route and give a hint to see, with a little bit of prompting, where the
student is in terms of her or his progress in making a particular mental construction.
This practice aligns with the paradigm, where the aim of an interview in particular
and of APOS-based research in general is not to organize students into categories
but to determine and explain how individuals construct their understanding of
mathematical concepts. Such an approach, that includes follow-up questions and
prompting, enables the interviewer to observe the construction process as it unfolds.
This can be thought of as an application of the notion of zone of proximal develop-ment introduced by Vygotsky (1978).
Once the instruments have been administered, the data is organized so that the
researchers can easily work with it. In the case of interviews, the audio and/or video
recordings are transcribed. Everything is carefully noted, including the sounds that
students make, intervals of silence, words that cannot be heard clearly, and gestures
(in the case of video recordings). The transcriptions are divided into segments of
related content, with descriptors used to indicate the general idea of that content.
Each member of the research team analyzes the transcriptions separately. They then
convene to discuss their findings and, if necessary, to negotiate differences in
interpretation until they reach consensus. This is a form of triangulation that has
proven to be effective in APOS-based research and is one of the main reasons why
the majority of published papers that use this paradigm have multiple authors. The
steps of interview analysis as a method are given in Asiala et al. (1996) as follows:
1. Script the transcript: The transcript is organized in a two-column format where
the original transcript appears in the first column in segments. The second
column contains a brief explanation of what happens in each segment.
2. Make the table of contents: Each item in this table summarizes one or more of
the explanations appearing in the second column.
3. List the issues: “By an issue we mean some very specific mathematical point, an
idea, a procedure, or a fact, for which an interviewee may or may not construct
an understanding. For example, in the context of group theory one issue might be
whether the student understands that a group is more than just a set, that is, it is a
set together with a binary operation” (Asiala et al. 1996, p. 27). Generally after
each researcher makes a list of issues, discussion and negotiation gives rise to a
single set of issues to be considered.
4. Relate to the theoretical perspective: At this point, interpretation of issues is
made through the lens of APOS Theory.
5. Summarize performance.
96 6 The APOS Paradigm for Research and Curriculum Development
Following are illustrations of this analysis.
Table 6.1, taken from the data obtained for the study reported in Dubinsky et al.
(2013), illustrates a portion of the first step, a scripted interview. An interview
extract appears in the first column, and the second column provides a summary/
brief explanation of the contents of the first column.
Table 6.2, taken from the same study, illustrates the relation between Step 2 (list
of issues) and Step 4 (relate to the theoretical perspective) for Anita, one of the
students who participated in the study. Column 1, denoted Entry, indicates the
Table 6.1 Scripting of an interview transcript
**********************************************
I: Alright so this represents one (base ten flat).
It happens to be split into a hundred equal pieces,
which makes one square 1/100. If I were going
to shade to represent .9 repeating, what would
I shade?
45. How much of base
10 flat ( ¼1) should
be shaded?
Anita: Well. Just pretty much everything down to there.
I: That tiny little bit.
Anita: Yeah.
I: Alright, so let’s say we’ve got that.
**********************************************
That tiny little bit though creates a gap again. How
do you see that gap?
46. 0.9(9) means “all the
nines,” so no gap;
0.9(9) ¼ 1.Anita: You know I think I guess the gap that I am seeing
in there is just adding another nine, but .9
repeating, you can’t add another nine, so I guess
it does equal 1, because we can’t add – yeah.
I: Ok. Repeat everything you just said.
Anita: Ok. I guess because what I was thinking when I just
said that, is that it could keep getting closer to
the entire one if we just added another nine.
But .9 repeating is essentially all the nines, I mean,
infinite. So, I guess, you can’t get in between there
so I guess it is one. I keep saying there is a little gap.
I: Yeah you do.
Anita: But there can’t be, because you can’t add just
another nine, because there is already all
the nines on there, right?
I: Yeah.
Anita: Ok.
**********************************************
I: But where did this come from? I mean how did
this come to light like this? That’s what I’m
curious about.
47. Process of shading led
to seeing all of the 9s
at once; accepts
statement D; no gap.Anita: Well, it just says that I was supposed to shade that
in, and I was thinking, well, it could go all the way
down to there and you add another nine and you
could just keep getting closer. But we’re not
adding nines, because all the nines are already
on there.
6.2 Data Collection and Analysis 97
Table 6.2 Issues for Anita
Entry Script Comment Issue Interpretation (relation to theory)
1 Meaning of the
symbol
0.35454. . .
IA: Process
ED: Process
KW: Process
R: Process
IA: Writing out a repeating decimal
ED: Sees process in repeating decimal a Little
weakly
KW: Appears to see process ion repeating
decimal
R: Appears to see process in repeating decimal
2 Fraction that
corresponds to
0.35454....?
IA: Correspondence;
influence of
authority
ED: Correspondence
KW: Correspon-
dence
R: Correspondence;
influence of
authority
IA: There is a corresponding fraction; they
learned a method to find it, which the
student does not remember
ED: A fraction corresponds to a repeating
decimal, but the student is a Little unsure
KW: The student believes there is a fraction
for the repeating decimal and some “for-
mula” to find it; the student does not recall
the formula
R: The student believes there is a fraction for
the repeating decimal and some “formula”
to find it but does not recall it
3 Item 10, written
instrument
IA: Approximation
ED: Object
KW: Object
R: Object
IA: Confusion: A repeating decimal is a defi-
nite number or an approximation?
ED: A repeating decimal is a definite number
KW: The student says a repeating decimal is a
“definite number of anything,” but her
indecision suggests that such a conception
may be tenuous
R: The student says that a repeating decimal is
a “definite number of anything,” but her
indecision suggests that such a conception
may be tenuous
4 ¼ � 0.25 or
¼ ¼ 0.25
IA: Action; approxi-
mation
ED: Decimals;
equality
KW: Equality;
approximation
R: Equality; approx-
imation; division
IA: Action of long division: ¼ ¼ 0.25, not an
approximation because the division
“worked out evenly”
ED: A finite decimal equals its fraction
KW: Student says that 0.25 is an approxima-
tion of ¼, but after doing long division, she
changes her mind
R: The student says that 0.25 is an approxi-
mation of ¼ but after doing the long divi-
sion changes her mind
5 0.333. . . anapproximation
of 1/3
IA: Action; approxi-
mation
ED: Equality
KW: Approxima-
tion; process
R: Approximation;
process
IA: Action of long division: 0.33. . . is but anapproximation of 1/3 because in the
conversion of 0.33. . . you do not get
exactly 1/3
ED: 1/3 is not exactly equal to 0.33. . .; thestudent is a Little uncertain
KW: The student sees long division of 1/3 as
a process in the sense that she understands
that the same remainder recurs over and
over. She claims that the quotient is only
an approximation
(continued)
98 6 The APOS Paradigm for Research and Curriculum Development
number of the interview segment. Column 2, labeled Script Comment, gives the
comment that accompanies the interview segment. Column 3 reports the issue(s) for
each interview segment. The initials “IA,” “ED,” and “KW” indicate one of the
researchers, and the word or phrase that appears after each initial is the issue(s)
the researcher attributes to the interview segment. “R” indicates the resolution or
the result of the researchers’ joint discussion and negotiation. Column 4 gives the
theoretical interpretation. Each researcher’s individual analysis appears after the
initials “IA,” “ED,” and “KW,” and the joint interpretation appears after “R”.
Summarizing the data for Anita (Step 5), she reached the Process stage but not
the Object stage. She made progress towards the latter and, in terms to be discussed
in Chap. 8, was considered to have reached the level of an emerging Totality
conception, which lies between Process and Object. Finally, regarding the equality
0:�9 ¼ 1, she tended towards disbelief, which is consistent with her difficulty in
seeing the infinite repeating decimal 0.999. . . as a mental Object.
In this study, and many other APOS studies, the steps of organization and
analysis described and illustrated in this section are often used to test the validity
of the genetic decomposition: if there is a mathematical issue that some students
appear to understand, but others have difficulty with, the researcher uses the data to
see if the difference can be explained in terms of the presence or absence of one or
more specific mental structures and/or relations between mental structures. If this is
the case, then the presence of these structures in the genetic decomposition is
supported. If this does not seem to be the case, then the researchers may need to
consider a revision of the genetic decomposition.
To illustrate the discussion in the above paragraph, an example from Cottrill
et al. (1996) will be revisited. In Sect. 4.5, a genetic decomposition for limit and its
refinement based on Cottrill et al. (1996) were presented. Table 6.3 showing both
genetic decompositions is repeated below for easy reference.
In this study, the need for refinement was revealed when data indicated that some
students did not appear to have been making some of the constructions proposed by
the genetic decomposition. For example, the interiorization of Step 2P of the
preliminary genetic decomposition was revised to include a three-step construction
of a Schema, as shown in Step 3R of the refined genetic decomposition.
Below is an interview excerpt from this study revealing the need to introduce
Step 3R(a) of the refined genetic decomposition.
Table 6.2 (continued)
Entry Script Comment Issue Interpretation (relation to theory)
R: The student sees long division of 1/3 as a
process in the sense that she understands
that the same remainder recurs over and
over. She believes that the quotient is only
an approximation
6.2 Data Collection and Analysis 99
Norton: Um, if you start, if, if a was, say, 3 and you started with x as 10, you go 10, 9, 7,
6, whatever, then you get 3.2, 3.1, and get smaller intervals between x and a, the functionwill approach a limit and the limit will be L.
I: . . . How close do things have to get before you’re willing to call L a limit?
Norton: Oh, tricky. Um. . . If, well, it should—you want get, start getting smaller intervals
between x and a and if you do a sequence of points, you will see it approaching a
limit L. If it doesn’t approach a certain number, then the limit doesn’t exist. But in this
function, it says that it has the limit L, so as your x approaches a, you should, the
function should assume the limit L.
In this excerpt, Norton explicitly selects values of x that approach a but does notdo this for values of the function. Although he mentions “it approaching a limit L,”which is a reference to values of the function, the focus is on the domain and on xapproaching a in isolation from the values assigned to each domain.
6.2.2 Written Questions
Written questions can be administered to large groups of students during an exam
or in the form of a questionnaire. They provide basic information on students’
Table 6.3 Preliminary and refined genetic decompositions of limit (based on Cottrill et al. 1996)
Preliminary genetic decomposition Refinement
1R: The Action of evaluating the function
f at a single point x that is consideredto be close to, or even equal to, a
1P: The Action of evaluating the function fat a few points, each successive point
closer to a
2R: The Action of evaluating the function
f at a few points, each successive point
closer to a
2P: Interiorization of the Action of Step
1P to a single Process in which f ðxÞapproaches L as x approaches a
3R: Construction of a coordinated Process Schema:
(a) Interiorization of the Action of Step 2R
to construct a domain Process in which
x approaches a.(b) Construction of a range Process in which
y approaches L.(c) Coordination of (a) and (b) via f
3P: Encapsulation of the Process of Step
2P so that the limit becomes an Object
to which Actions can be applied
4R: Encapsulation of the Process of Step 3R(c)
so that the limit becomes an Object to which
Actions can be applied
4P: Reconstruction of the Process of Step
2P in terms of intervals and inequalities.
This is done by introducing numerical
estimates of the closeness approach:
0 < x� aj j < δ and 0 < f ðxÞ � Lj j < ε
5R: Reconstruction of the Process of Step 3R(c)
in terms of intervals and inequalities. This is
done by introducing numerical estimates of
the closeness approach: 0 < x� aj j < δ and0 < f ðxÞ � Lj j < ε
5P: Application of a two-level quantification
Schema to connect the Process described
in Step 4P to the formal definition
6R: Application of a two-level quantification
Schema to connect the Process described in
Step 5R to the formal definition
6P: Application of a completed ε� δconception to specific situations
7P: Application of a completed ε� δ conceptionto specific situations
100 6 The APOS Paradigm for Research and Curriculum Development
mathematical performance. They can also be used in the design of interview
questions because of their ability to reveal student difficulties that require further
analysis.
Asiala et al. (1997b), in a study of students’ learning of the concepts of cosets,
normality, and quotient groups, illustrate how a genetic decomposition informs the
design of written instrument questions. Their analysis was guided by the prelimi-
nary genetic decomposition developed by Dubinsky et al. (1994). This decomposi-
tion is simply the progression Action to Process to Object. The three stages can be
described as follows:
An action conception of coset has to do with forming a coset in a familiar situation where
formulas or recipes can be used such as the multiples of 3 in Z or in Z18. The action
conception is not strong enough to handle formation of cosets in more complicated
situations such as in Sn beyond the familiar examples of S3 or S4, where cosets are not
generally representable by formulas or simple recipes.A process conception of coset will allow an individual to think of the (left) coset of a
subgroup by an element by imagining the product of that element with every number of the
subgroup – without having to actually form the products. In a process conception of coset,
the main thing one thinks about is the formation of the coset.With an object conception of coset, an individual can think about, name and manipulate
a coset without necessarily focusing on how it is formed. Actions can be performed on
cosets such as comparing the cardinality of two cosets or counting the number of cosets,
both of which arise in the proof of Lagrange’s theorem. . .There are other actions which can be applied to cosets as objects, such as considering
the relations among elements, subgroups and cosets expressed by the various equivalent
conditions for normality or the formation of product of cosets. (Asiala et al. 1997b,
pp. 247–248)
Students in two introductory abstract algebra classes participated in the study.
One class followed an APOS-based instructional strategy using the ACE cycle, and
the other class followed a traditional approach.
The research instruments included two written exams and interviews. These
tools were used to determine whether students made the mental constructions called
for by the preliminary genetic decomposition. Students worked on the first exam in
their cooperative groups and completed the second one individually.
In APOS-based research, written questions are carefully designed to help gather
evidence for the presence of the mental constructions predicted by the preliminary
genetic decomposition and to suggest modifications of the pedagogical strategies
and/or the genetic decomposition when these constructions are not present. They
also allow researchers to focus their attention on the aspects of the construction of
knowledge that they are studying. Some examples of these questions related to
Action, Process, and Object conceptions in abstract algebra, taken from the Asiala
et al. (1997b) study, follow.
Responses to Question 5 from Test 1 provided evidence of the construction of
specific actions:
Find a subgroup of S4 that is the same as S4. Calculate the left cosets of your subgroup.
(Asiala et al. 1997b, p. 305)
6.2 Data Collection and Analysis 101
In this question, once the subgroup is specified, an Action conception of coset
consists of listing the elements of each coset. Here specific cosets of a particular
subgroup which is familiar to the student are to be formed. The student can do this
by applying a formula, which is an external cue.
Responses to Question 4(a) from Test 2 were used to determine the construction
of Process conceptions:
There are many conditions that are equivalent to a subgroup H of a group G being normal.
One is,
for all g 2 G; it is the case that gHg�1 � H
Give another condition for normality and show that it is equivalent to this statement. (Asiala
et al. 1997b, p. 306)
The answer to this question may reveal how the student “thinks of a coset and
also perhaps the way in which manipulations with cosets are performed” (p. 306).
Ability to use set former notation in formation of cosets, such as in fgh : h 2 Gg;was considered to be an indicator of Process, since this provides evidence that the
student can think about forming the cosets in her or his mind, without actually
having to form them. The generality of the question, without the mention of a
specific group or a specific coset provides an appropriate context to check for
interiorization of Actions.
Responses to Question 6 from Test 2 were used to determine whether encapsu-
lation had occurred (Asiala et al. 1997b, p. 307):
Let S3 be the group of permutations of three objects.
(a) Find a normal subgroup N of S3.(b) Identify the quotient group S3=N.
Finding the subgroup of a given group is an Action applied to that group.
Similarly forming a quotient group is an Action applied to the group. Identification
of a quotient group is an Action applied to the quotient, which is itself an Action
applied to a set of cosets. The ability to apply such Actions indicates an Object
conception of coset.
6.2.3 Classroom Observations
Classroom observations can reveal interesting data, especially when the instruction
is not based on pedagogical elements related to APOS Theory or is implemented by
instructors with little or no experience with this approach.
Ku et al. (2008) observed an introductory one semester “linear algebra for
engineers” course with 24 students. Although the course was designed according
to elements of APOS Theory, the classroom observation brought to light the fact
that there were several aspects of an APOS-based pedagogical strategy that were
102 6 The APOS Paradigm for Research and Curriculum Development
missing from the course design. For example, while the computer activities were
done with a computer algebra system, the emphasis was put on its use as a tool for
calculating answers instead of programming with the purpose of facilitating mental
constructions. None of the students showed either Process or Object conceptions of
the concept of basis of a vector space. This may be attributed to the fact that the
pedagogy did not encourage these types of mental constructions.
6.2.4 Textbook Analyses
Textbooks can be analyzed to account for the pedagogical strategy followed, to
determine which results, rules, and theorems make use of the concept, and to
investigate the notation employed that might have bearing on students’ understand-
ing. All these elements can be useful when analyzing student responses and can be
used to inform the interpretation of data. Roa-Fuentes and Oktac (2010) performed
such an analysis for the linear transformation concept. They proposed two genetic
decompositions for the concept of linear transformation. In their analysis of student
data, they found evidence for one of the genetic decompositions but not the other.
Their analysis of linear algebra textbooks uncovered the reason. As explained in
Chap. 4, one of the preliminary genetic decompositions that they proposed started
with a previously constructed (general) transformation Object and built on that by
de-encapsulating it and coordinating the underlying Process with the related binary
operation Processes. On the other hand, the textbook used in the course, as well as
the instructor, defined a linear transformation as a function between vector spaces;
neither of them introduced the general transformation concept before that. There-
fore, it was unsurprising to find no empirical evidence for the general transforma-
tion concept. Thus, it is necessary to implement an instructional approach based
on the idea proposed in this genetic decomposition in order to test whether the
linear transformation concept can be constructed in the way not considered in the
textbooks.
6.2.5 Historical/Epistemological Analysis
For studies of students’ conceptions of mathematical infinity (e.g., Dubinsky et al.
2005a, b), historical/epistemological analyses were conducted. The study of the
historical development of mathematical infinity helped in contextualizing student
difficulties in terms of obstacles that mathematicians faced throughout the develop-
ment of this mathematical concept as well as in explaining these difficulties in
cognitive terms. For example, Weller et al. (2004) describe “how a particular theory
about how people came to understand mathematics, APOS Theory, can be helpful
in understanding the thinking of both novices and practitioners as they grapple with
6.2 Data Collection and Analysis 103
the notion of infinity” (p. 741). In that article and in Dubinsky et al. (2005a, b),
several issues and paradoxes related to the mathematical development of the
concept of infinity are analyzed. These include the paradoxes of Achilles and
the Tortoise and analogues of Hilbert’s Hotel, infinitesimals, the relation between
the union of all finite segments f1; 2; . . . ; ng of natural numbers and the set of all
natural numbers, and the possibility of using a countable mental procedure to obtain
the mental construction of an uncountable set. Underlying most of these issues is
the distinction between potential infinity and actual infinity. This latter issue has
puzzled mathematicians and philosophers for millennia, from Aristotle to at least
Poincare in the beginning of the twentieth century. In these articles, the authors
argue that the paradoxes might be resolved and the issues might be settled in the
minds of the learners by an analysis that considers examples of potential infinity to
be Processes and examples of actual infinity to be the mental Objects obtained by
encapsulating those Processes. They go on, in several studies (Weller et al. 2009,
2011; Dubinsky et al. 2013), to use genetic decompositions based on this analysis to
provide empirical data in support of their argument.
6.3 Types of APOS-Based Research Studies
Weller et al. (2003) report on student performance and attitudes in courses based on
APOS Theory. They classify research studies into four types:
• Comparative studies in which the performance of students who received instruc-
tion using APOS Theory and the ACE Teaching Cycle is compared with the
mathematical performance of students who completed traditional lecture/recita-
tion courses;
• Non-comparative studies measuring the performance of students who completed
courses using APOS Theory and the ACE Teaching Cycle;
• Studies of the level of cognitive development of students who completed courses
based on APOS Theory and the ACE Teaching Cycle or courses using a
traditional lecture/recitation model;
• Comparisons of student attitudes and the long-term impact of courses based
upon APOS Theory and the ACE Teaching Cycle to that of students who
completed traditional lecture/recitation courses. (p. 98)
A study may fall into more than one category depending on the type of data used.
The study conducted by Asiala et al. (1997b) is an example that falls into the first
three categories. Explanations and examples of all four categories follow.
104 6 The APOS Paradigm for Research and Curriculum Development
6.3.1 Comparative Studies
The first category corresponds to studies where students who received instruction
other than an APOS-related approach were interviewed for comparative purposes.
This gives a general idea of how successful (or not) the APOS pedagogical strategy
was in helping students to construct their understanding of a specific mathematical
concept. This was the case with Asiala et al. (1997b), where the students who
learned about cosets, normality, and quotient groups following the pedagogical
approach based on APOS Theory performed better than the group that followed a
traditional approach.
Also belonging to this category is the longitudinal study reported in Weller et al.
(2009, 2011). There were several factors that led the researchers to adopt a
comparative analysis. The fact that there were few studies on experimental
approaches on the topic suggested the need for a comparative analysis, and the
large number of student participants made such a study possible. Weller et al.
(2009) found that preservice elementary and middle school teachers who completed
a specially designed unit on repeating decimals based on APOS Theory and
implemented using the ACE teaching cycle made considerably more progress in
their development of an understanding of the equality 0:�9 ¼ 1 and the more general
relation between a rational number and its decimal expansion(s) than their control
group counterparts (see Chap. 8 for details regarding the study and Chap. 5 for
information about the instructional design). A second study (Weller et al. 2011),
based on interviews conducted four months after the instruction, reports that the
students “who received the APOS-based instruction developed stronger and more
stable (over time) beliefs that a repeating decimal is a number; a repeating decimal
has a fraction or integer to which it corresponds; a repeating decimal in general
equals its corresponding fraction or integer; and, in particular, 0:�9 ¼ 1” (p. 129).
6.3.2 Non-comparative Studies
The study conducted by Asiala et al. (1997b) also corresponds to the second
category, since non-comparative as well as comparative data were used. The
non-comparative data came from exams, as described in Sect. 6.2.2, that were
administered only to those students in the experimental course. Two sets of
interviews were conducted, the first one providing non-comparative data that was
carried out only for the students who followed the APOS-based course.
6.3 Types of APOS-Based Research Studies 105
6.3.3 Studies of the Level of Cognitive Development
The study conducted by Asiala et al. (1997b) is also considered as belonging to the
third category because it gives a detailed account of how students construct their
understanding of certain abstract algebra concepts, namely, cosets, normality, and
quotient groups.
Other studies belonging to the third category are those where the goal was to
observe the viability of a preliminary genetic decomposition in the absence of
previous studies about the learning of a particular mathematical topic or of a
specifically designed APOS-related instructional sequence. As explained in
Chap. 4, Roa-Fuentes and Oktac (2012) have conducted such interviews for the
purpose of looking for evidence for one or both of the preliminary genetic
decompositions that they had proposed for the linear transformation concept
(Roa-Fuentes and Oktac 2010). They found evidence for only one of them,
commenting that this may be due to the kind of instructional treatment that the
students had received.
In another study about functions of two variables, Trigueros and Martınez-
Planell (2010) designed an interview in order to find information about the
components of the preliminary genetic decomposition that they proposed. In their
study, they focused on the analysis of students’ responses to those questions related
to subsets of R3 and graphs of functions of two variables. They conducted nine
interviews after the students had finished a course on multivariate calculus for
undergraduate mathematics students. In their preliminary decomposition, they
suggested that the construction of a Schema for R3 and the Processes involved in
drawing graphs of functions of one variable were prerequisites for the learning of
the concept of functions of two variables. In the preliminary genetic decomposition,
a Process to construct fundamental planes was included.
Results of the study showed that most of the interviewed students had indeed
constructed a Schema for R3 as predicted. However, their Schema included neither
the construction of subsets of R3 nor the coordination of the Schema for R3 and that
of a function of one variable. Most students in this study were not able to perform
Actions on any Object in space but points. The researchers concluded that it was
necessary for students to construct subsets of points in space as Objects as well as to
coordinate between the Schema for R3 and that for functions of one variable in
order to be able to construct the concept of function of two variables. Comparison
of results achieved by different students showed that the Action of intersecting
surfaces with planes and the interiorization of this Action into a Process in which
the result of the intersection can be predicted played an important role in students’
learning of these functions. Since these constructions were not predicted in the
preliminary genetic decomposition, they were introduced in its refinement.
The refined genetic decomposition was tested in a second study (Martınez-
Planell and Trigueros 2012) where results showed that it was a good model
of students’ mental constructions. This genetic decomposition has been tested
in instruction.
106 6 The APOS Paradigm for Research and Curriculum Development
6.3.4 Comparisons of Student Attitudes and the Long-TermImpact of APOS-Based Instruction
The fourth category consists of studies which focused attention on students’
attitudes. The purpose of these studies was to investigate the effect of APOS-
based pedagogical strategies on students’ attitudes when compared with the
attitudes of students who completed learning units based on other types of instruc-
tion. Weller et al. (2003) summarized the results of a study on student attitudes in
abstract algebra in which the researchers found that students who completed an
APOS-based experimental course felt more positive about mathematics than those
who completed a traditional course covering similar content. However, they quali-
fied these results, noting that factors related to the instructors, the interviewers, and
the grade distribution may have affected the findings. They also pointed out that no
attempt was made to conduct a statistical analysis.
6.4 Scope and Limitations of APOS-Based Research
APOS is a cognitively oriented theory and as such provides a useful tool for
modeling student understanding of mathematical concepts. It also has a social
component that relies on cooperative learning, as the context of group work is
more likely to give rise to more explicit questions, doubts, and explanations by
students than what would typically transpire in individual contexts (Vidakovic
1993). Moreover it provides a context that facilitates learning. That is, APOS
Theory functions under the premise that working in groups makes a difference in
the affective domains of the individuals.
APOS Theory has been successful in proposing models to explain the learning of
numerous mathematical concepts, including those which pose serious difficulties
for students such as linear independence, quotient groups, functions, and repeating
decimals. The literature does not mention very many (if any) other such success
stories about the learning of these topics.
The questions that can be asked in research studies where APOS Theory is used
as a tool for analysis are generally of the following types:
• How might an understanding of the concept be constructed by students?
• What are the mental constructions involved in the development of a Schema and
its components?
Some auxiliary questions can be used to help in answering the main questions or
can aid in making suggestions for didactical approaches based on research.
Examples are the following:
• What are the prerequisite concepts necessary to construct understanding of
a particular mathematical concept?
6.4 Scope and Limitations of APOS-Based Research 107
• How is a particular conception characterized in the learning of a particular
concept?
• How is the transition from one conception to another characterized?
• What are some pedagogical strategies that can help students in the mental
construction of a particular concept?
This does not mean that the research questions have to follow a specific format if
APOS Theory is used but rather emphasizes the kind of phenomena that arise in
working with this paradigm. The following are examples of topics that could be
researched using APOS Theory:
• The use of APOS Theory to find out how mathematicians perceive the mathe-
matical concepts they teach (see Stenger et al. 2008).
• Strength and stability of constructions of mathematical concepts learned using
an APOS-based learning sequence (see Weller et al. 2011).
• How to teach APOS Theory to pre or in-service teachers, so as to help them to
use APOS Theory methodology in their practice.
• Is APOS Theory applicable/adaptable to other humanistic or scientific domains?
The research trend using APOS Theory indicates that the construction of mathe-
matical concepts will continue to be studied using the lens of APOS Theory and
more research questions will be added to the repertoire as researchers discover the
need for it.
108 6 The APOS Paradigm for Research and Curriculum Development
Chapter 7
Schemas, Their Development and Interaction
APOS Theory has been successful in describing and predicting the types of mental
structures students need to construct in order to learn abstract concepts. As new
research is carried out and complex research projects are undertaken, it has become
necessary to widen the scope of the theory. This has been achieved by expanding
the researchers’ understanding of various theoretical constructs. Although there has
been less research using these constructs, they already form part of the theory or are
being tested in current research. One of these constructs is Schema; another is the
mechanism of thematization and another, to be discussed in Chap. 8, is a possible
new stage, Totality, between Process and Object.
Schema is the central subject of this chapter. A general description appears in
Sect. 7.1. Several examples are considered in Sect. 7.2. How Schemas develop is
the subject of Sects. 7.3 and 7.4. Sections 7.5 and 7.6 detail how a Schema changes,
either through the introduction of new information or through the interaction of two
Schemas. Finally, Sect. 7.7 deals with the issue of thematization, the mechanism
involved in the construction of an Object from a Schema.
7.1 Schemas in Piaget’s Work and in APOS Theory
Piaget and Inhelder (1966/1969) relate formal schemas to “. . . the concepts whichthe subject potentially can organize from the beginning of the formal level when
faced with certain kinds of data, but which are not manifest outside these
conditions” (p. 398). That is, schemas are related to important ways of reasoning
and refer to structures brought to bear on certain learning situations. Piaget consid-
ered schemas as a way to work with classes of situations in order to make sense of
them and to achieve various goals.
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_7,© Springer Science+Business Media New York 2014
109
For Piaget, schemas are instruments that structure knowledge:
A schema is only constructed when it is functioning, and it only functions through
experience: then, that which is essential is, not the schema as structure in itself but the
structuring activity that gives rise to schemas. (Piaget 1975/1985)
Any particular schema in itself does not have a logical component, but Schemas
are coordinated with each other, and this fact results in the general coordination of
actions. These coordinations form a logic of actions that is the beginning of logico-
mathematical structures. Piaget said that a schema could include subschemas or
subsystems. The subschemas are included in the total Schema in the same way that
a logico-mathematical structure of classification into subclasses is included within
the whole class. At a later stage, this relationship of class inclusion gives rise to
certain concepts (Piaget 1975/1985).
Piaget’s ideas are reflected in APOS Theory. Asiala et al. (1996) describe an
individual’s Schema for amathematical topic as all of her or his knowledge connected
(explicitly or implicitly) to that topic. Specifically, an individual’s Schema for a
certain mathematical concept is the individual’s collection of Actions, Processes,
Objects, and other Schemas which are linked by some general principles or relations
to form a framework in the individual’s mind that may be brought to bear upon a
problem situation involving that concept (see Sect. 3.2.5). This framework must be
coherent in the sense that it gives, explicitly or implicitly, a means of determining
which phenomena are in the scope of the Schema and which are not (Dubinsky
and McDonald 2001). A Schema can be thematized to become a cognitive Object to
which Actions and Processes can be applied. By consciously de-thematizing a
Schema, it is possible to obtain the original Actions, Processes, Objects, and
other Schemas from which the Schema was constructed (Clark et al. 1997).
A particular Schema may not necessarily be accessed in all situations, because
mathematical learning is highly nonlinear. However, the structure of a Schema and
its development may explain why students have difficulty with different aspects of a
topic and may even have different difficulties with the same situation in different
encounters.
Schema has been part of APOS Theory since its beginning (see Chap. 2 for
further details). In earlier papers, it was only referred to as one of the possible
constructions in the theory, but no research was conducted in which Schema had a
central role. As APOS-based research progressed, it was found that the Schema
structure was necessary in order to describe certain learning situations such as those
that will be described in Sect. 7.4.
7.2 Examples of Schemas
Some examples may be helpful to understand Schemas in APOS Theory. In each
example given below, the constructions included in the Schema are described. It is
important to note that the structure of a Schema can differ among different
individuals because each individual constructs different kinds of relations among
110 7 Schemas, Their Development and Interaction
the components of a Schema. Some of these differences are discussed in this section
and the sections that follow.
One example of a Schema is the function Schema. It can be composed of
different types of functions such as real-valued functions, multivariable functions,
vector-valued functions, and/or proposition-valued functions. These different types
of functions may have been constructed as Processes or Objects, together with the
operations that can be applied to them. For some students, different types of
functions may be related by the common idea of a function Process: an operation
applied to a set of inputs to obtain a set of outputs. Functions differ in the types
of inputs involved, the nature of the operations applied to those inputs, and the
results of the operations. Although individuals’ Schemas may include the same
types of functions, their components or the types relations constructed among them
may differ.
Another example of a Schema is the vector space Schema. It is composed of
vectors and operations defined on them, together with linear combinations, span-
ning sets, bases, and dimension, each considered as a Process or an Object
(Parraguez and Oktac 2010). For some individuals, these concepts are related to
the concept of vector space only because they might be defined for vector spaces.
However, other individuals who have constructed a coherent Schema are aware of
the nature of their relation to the vector space concept, for example, whether every
combination of vectors in a vector space is linear.
A third example of a Schema is the Cartesian plane Schema. It is a structure that
includes points as Objects and relations among points, such as curves, functions,
and regions that are constructed as Processes that result from interiorization of
the Actions of representing their points. For some individuals, these relations may
consist only of specific Processes applied to points to obtain new curves, functions,
or regions. For others, these relations may include the distinction between
different sets of points through their definition. A different Schema of the Cartesian
plane, constructed for the concept of equivalence classes of fractions, is described
in Sect. 9.5.
A Schema can be considered as being composed of different components. It
may include a single concept that can be applied to different situations, as in the
case of function, or it may include different but interrelated concepts, as in the
case of the vector space Schema. In either case, a Schema is a tool for under-
standing how knowledge is structured and its development through the learning
process.
As an individual’s mathematics learning progresses, different concepts need to
be related and used in problem-solving activities. Sometimes new Actions, Pro-
cesses, or Objects can be assimilated to a previously constructed Schema by
establishing new relations among the components of the Schema. In other
situations, a Schema may be related to one or more different Schemas that lead to
the construction of a new, more extensive Schema. For example, the Schema for the
Cartesian plane can be related to other higher dimensional spaces that results in the
construction of a new Schema that includes n-dimensional spaces and also
non-Cartesian spaces.
7.2 Examples of Schemas 111
When a person confronts a mathematical problem situation, he or she evokes a
Schema and makes use of some of its components and some relations among them
to deal with the situation. When facing the same situation, different persons may
use the same components but construct different relations among them. The study
of the relations together with the type of constructions brought to bear in dealing
with a particular problem-solving situation reveals the structure of an individual’s
Schema and gives information about its development. In APOS studies of the
development of individual Schemas, researchers have used “the triad,” a progres-
sion of three stages proposed by Piaget and Garcıa (1983/1989). The mechanism of
accommodation accounts for the progression from one stage of the triad to the next.
Each subsequent stage of the triad involves the development of relations and
transformations an individual can make between particular constructs within the
schema as well as the development of the coherence of the Schema in terms of its
possible application to specific problem situations.
In APOS-based research, the triad progression of stages has been used to
describe the development of students’ Schemas associated with specific mathemat-
ical topics and to better understand how Schemas are thematized to become
cognitive Objects. Schema development has proven to be an effective way to
understand this facet of cognitive construction and has led to a deep understanding
of the construction of Schemas (Trigueros 2005).
7.3 Development of a Schema in the Mind of an Individual
From the beginning, the use of Schemas in APOS Theory required the introduction
of the notion of Schema development; this development is described in stages.
Piaget and Garcıa call the stages involved in the development of any schema Intra-,
Inter-, and Trans-. The hyphen symbol “-” is followed by the name of the Schema
being discussed to indicate its application to a particular Schema. Since Schemas
can be defined by structures that differ in their complexity, this terminology helps to
distinguish the Schema described by the three stages. For example, a Schema for
functions includes different types of functions, such as in the example described in
Sect. 7.2, or it might include all the Objects and relations in differential and integral
calculus. Both examples are function Schemas whose development can be
described by stages, for example, the first one could be named Intra-function,
Inter-function, and Trans-function, and the second Intra-calculus, Inter-calculus,
and Trans-calculus in order to distinguish what is comprised by the Schema. It
would be possible to use function to name the stages of the second schema, together
with acknowledgment of its complexity level by making its components explicit.
The first stage, Intra-, is marked by a focus on individual components of a
Schema. This consists mainly of correspondences among the system components.
The individual discovers a set of properties that are common among the Objects that
are included in the Schema, with all connections being local and particular. For
example, in the historical development of geometric structures, the stages of
112 7 Schemas, Their Development and Interaction
Schema development can be described in terms of the relationships between figures
and space. In the Intra-figural stage, the study of geometric figures in Euclidean
Geometry focuses on the representation of figures and the study of their properties.
It is considered that figures are part of space, but space is regarded as a global entity
where figures can be studied. In this stage, it is possible, for example, to establish a
correspondence between a segment and a number to define a unit of measurement.
This kind of correspondence is an internal relation between the elements of the
figure (Piaget and Garcıa 1983/1989).
As knowledge develops, comprehension of local transformations starts to play a
more fundamental role and access to necessary connections and the reasoning
behind them begins to be developed. At this point, the Schema is said to be at the
Inter-stage. In the case of geometrical structures, algebraic representations in
Analytic and Projective Geometry lead to the introduction of a system of
transformations; these transformations relate the figures under different perspectives.
This is characteristic of the Inter-figural stage (Piaget and Garcıa 1983/1989).
Later on, it is necessary to determine the links and reasons behind the local
transformations that constitute the Schema. In particular, an individual begins to see
the Schema as a whole, and a structure that can account for its composition as a
whole is constructed by means of synthesis. The structure is now coherent, and the
individual can determine whether it is applicable or not to a given situation. This is
called the Trans-stage. In the case of the development of the geometrical structures,
this stage is constructed when groups of transformations are introduced (Piaget and
Garcıa 1983/1989).
According to Piaget and Garcıa (1983/1989):
Passing from one stage to the other is not characterized by a period of “increments” in
knowledge with respect to the previous stages, instead, a total reinterpretation of the
conceptual fundaments is involved. . . access to the next stage needs the reconstruction of
what had been constructed in previous stages. (p. 109)
Piaget and Garcıa assert that all Schemas develop through a progression or series
of stages they called the triad. They hypothesized that these stages can be found
when analyzing any developing Schema. An important issue that needs to be
clarified is that for Piaget the nature of the triad stages was functional, not
structural. The focus on Schema in Piaget’s genetic epistemology is not the
structure of the Schema but the way Schemas function in cognitive development.
Central mechanisms of Piaget’s Theory, such as assimilation, accommodation, and
equilibration, play a fundamental role: the incorporation of new elements to the
Schema by assimilation leads to its modification through accommodation. It is
through these mechanisms that the Schema reaches a new equilibrium. Equilibrium
is dynamic, so that through it the Schemas are constantly changing, although they
maintain their identity.
The triad was first used in APOS Theory by Clark et al. (1997) in a study of
students’ understanding of the chain rule. It was also used in other studies such as
sequences of numbers (McDonald et al. 2000), the chain rule and its relation to
composition of functions (Cottrill 1999), and the relation between the graph of a
7.3 Development of a Schema in the Mind of an Individual 113
function and properties of its first and second derivatives (Baker et al. 2000; Cooley
et al. 2007). In all of these studies, the triad helped the researchers to understand the
development of Schemas and support their explanations of students’ thinking that
arose in the analysis of the data.
In APOS Theory, in line with Piaget and Garcıa, the development of the Schema
consists of three stages, Intra-, Inter-, and Trans-, and the triad progression of
stages is involved in the transition from one stage to the next through the develop-
ment of relations and transformations that an individual makes among the
particular constructs within the Schema: the Intra-stage of Schema development
is characterized by a focus on individual Actions, Processes, and Objects in
isolation from other cognitive items of a similar nature; the Inter-stage is
characterized by the construction of relationships and transformations among the
cognitive structures that make up the Schema where an individual may begin to
group items together and even call them by the same name; at the Trans-stage, the
individual constructs an implicit or explicit underlying structure through which the
relationships developed in the Inter-stage are understood and by which the Schema
achieves coherence that is indicated by the individual’s ability to determine what is
in the scope of the Schema and what is not (Dubinsky and McDonald 2001).
7.4 Examples of Development of a Schema
In this section, examples of Schema development at each of the triad stages are
discussed.
7.4.1 The Intra-Stage
In APOS Theory, the Intra-stage of Schema development is characterized by a
focus on individual Actions, Processes, and Objects in isolation from other cogni-
tive items. At the Intra-stage, the student concentrates on a repeatable action or
operation and may recognize some relationships or transformations among Actions
on different components of the Schema. Some examples may help to better under-
stand this stage:
Functions In the case of the function concept, an individual at the Intra-stage tends
to focus on a single type of function and the various activities that he or she could
perform with it (Dubinsky and McDonald 2001).
Derivative At the Intra-stage of the derivative Schema, the student can interpret
the derivative as the slope of a tangent line at specific points and can performActions
or Processes to find the derivative of a function on specific intervals. The individual
can determine if the derivative is positive or negative and use the sign of the
derivative to decide whether the function increases or decreases on those intervals.
114 7 Schemas, Their Development and Interaction
The student can also solve some rate-of-change problems. However, the focus is on
individual Actions or Processes, so the description of the behavior of a curve and the
solution of rate-of-change problems are only related by the need to find the deriva-
tive of a function. The individual is unable to consider the derivative as a means to
describe local variation of the function (Baker et al. 2000).
Chain Rule At the intra-stage of the chain rule Schema, the student has a collec-
tion of rules to calculate some individual cases where the chain rule is used
implicitly, such as the power rule or the exponential rule, but does not see any
relationship between those cases. The student considers each case as different rules
that can be applied to specific situations (Clark et al. 1997; Cottrill 1999).
The following is an example of a student at the Intra-chain rule stage.1 In this
case, Tim could see the chain rule for implicit differentiation but could not recognize
how it might be used in other problems where it applies (Cottrill 1999, p. 39):
I: Could you write down the chain rule in whatever words or symbols you remember?
Tim: The chain. . . ?
I: The chain rule, for taking derivatives.
Tim: You mean y ¼ x2 � 7xþ 5 and you take the chain rule for this one?
I: No, the chain rule is just a rule we have for taking certain kinds of derivatives. Do you
remember using the chain rule?
Tim: Um, y, it was something, um. . . Can I do an example out of the book? The chain rule for
this one? [writes out f ðxÞ ¼ yn; yn 0 ¼ nyðn�1Þy0. . .I: Let me show you an example of a problem where you would use the chain rule, for
instance #18 here. The original problem was sinð5x4Þ. [Student is shown written work
from the questionnaire, which has the correct solution to the problem].
Tim: You mean give the rules of this one, how I worked out this one?
I: Yes, how did you find the derivative there?
Tim: Oh, [writes and crosses out f ðxÞ ¼ uðxÞ0f 0ðxÞ [mumbles]. . . I don’t know, you know, if yougive me an example of how to do the chain rule, I know how to do products.
I: That is what I am saying, this solution you have right here. . .
Tim: Mm-hmm
I: you used the chain rule
Tim: yeah
I: to get that solution, which is correct. OK? Does this remind you of the chain rule, then?
Tim: Uh-huh, so what you want me to . . .
I: We are starting with sinð5x4Þ and look at what you wrote down for your answer, and try
and remember how you came up with that idea.
Tim: Oh! So first I take derivative of outside, derivative of sine is cosine
I: Right
Tim: so then I take derivative of inside, so inside is 5x4, so I write down 20x3.
I: OK
Tim: That is so easy, you know, I don’t know how to get the something that you asked me to do.
I: You don’t. . . So the question was, what does the chain rule say?
Tim: If there is a function of x, something like that, and take the derivative of that, right, first
you take the derivative of outside first, then take derivative of inside.
1 Problems for the excerpts of students’ responses shown in this section appear in the Appendix at
the end of this chapter.
7.4 Examples of Development of a Schema 115
Although Tim can remember and use correctly the power rule and, with some
help, use the formula for the chain rule, he describes his work in terms of Actions
related to specific problems. He describes each Action in isolation as if each rule
were different.
7.4.2 The Inter-stage
As mentioned in Sect. 7.3, the Inter-stage is characterized by the construction of
relationships and transformations among the Processes and Objects that make up
the Schema. At this stage, an individual may begin to group items together and even
call them by the same name. The Inter-stage is described for the examples consid-
ered in Sect. 7.4.1:
Function As an individual considers possible analogies among operations on
different types of functions, as Processes, he or she may construct a relation
among them and recognize types of functions as instances of the same sort of
activity, for example, as a means of constructing new functions from known ones.
Derivative At the Inter-stage of the derivative Schema, the student can relate the
Process of the derivative as the slope of the tangent line and the Process of the
derivative as the rate of change at a given point so that he or she can consider the
derivative as a means to describe local variation of the function (Baker et al. 2000).
Chain Rule In the case of the chain rule Schema, the Inter-stage is characterized
by recognition that different instances of the chain rule, such as the power rule or
the derivative of a composition of functions, represent something more general.
That is, the individual becomes aware that special cases are related and that those
rules are instantiations of a more general rule (Clark et al. 1997; Cottrill 1999).
In the next excerpt, Peg demonstrates an understanding at the Inter-chain rule
stage. She groups different differentiation problems, which include problems
involving compositions of functions, according to the chain rule as an initial
criterion. Although she distinguishes between exponential and trigonometric
functions, she keeps the chain rule as her main criterion. However, she is unable to
include in her criterion exponential or trigonometric functions applied to the identity
function because expressions for these functions do not include parentheses. She
fails to recognize implicit functions as instances where she needs to use the chain
rule, and she also fails to recognize the case where composition of functions is
used in an integral. It seems that her criterion for grouping is based on the use of
parentheses (Cottrill 1999, p. 46).
I: So is that five different groups?
Peg: [pause] Yes. Five different groups.
I: OK, what was the discriminating features?
Peg: OK, One is like the most straight forward, where it is just strictly using the uh, what’s the
name of that rule? The power rule.
I: Uh-huh.
116 7 Schemas, Their Development and Interaction
Peg: It’s just straight out the power rule. Uh, two and three are pretty much exactly the same
thing except for two, the expression needs to be rewritten to use that rule. 6 and 7 is also
just the power rule except for you have uh, uh, you have a power rule and then it’s a chain
rule. Uh, 4, 5, 8, & 9 all have trigonometric uh terms in them to where then you also have to
know their trig functions and then 10 is the e function which also has it’s separate rule.
I: OK. When you went through that. . .
Peg: And that’s pretty much from my point of view which was from the easiest to the most
difficult. Although, I guess that the e function is not really that difficult; it’s just knowing
that it, it’s just not as, it’s not difficult it’s just different because it just doesn’t seem to
follow the format as all of the others.
I: Uh-huh. Um, was there any other way that you considered grouping these?
Peg: [pause] Uh. . . in the time frame, I just went with the first thought that popped into my
mind. I’m sure that if I was to sit here longer, I could think of other ways to do it.
I: OK. What might be other criteria?
Peg: [pause] Uh, the other ones would have been, may be, the ones that just had single terms as
opposed to having two terms that have to be differentiated. Or anything that that has a
chain, or pulling out anything that has the chain rule and knowing that also have to add
more terms to the final expression.
I: So the last couple of questions that I have for you, write down the chain rule using
whatever words or symbols that you like. It’s as best as you can remember it. You have
here like examples of it. Number 6 and 7 which is16 and 17 on our papers here. You used
the chain rule in your answer. So if you want to use those to help spark some ideas. . .
Peg: [pause] [mumbles to self] [pause] OK.
I: Read that for the tape.
Peg: OK. Uh, when you have the derivative of f ðgðxÞÞ it’s f 0ðgðxÞÞ times g0ðxÞ. It’s basically youtake a derivative of the outer term and I’m using outer term because it’s just the way that
I look at it, composed with the inner term and then multiply it by the derivative of the
inner term.
I: OK. Great. Um, actually the last five statements here on the list, 6 through 10
Peg: Uh-huh.
I: were intended to be, to use the chain rule and you named 6, 7, & 8 as using the chain rule.
Do you see the chain rule being used in 9 and 10?
Peg: Yes, I do, but I had pulled them out separately because they had the trigonometric
functions also.
I: OK, no that’s fine. Does your rule that you have written there at the bottom, does it apply
or does it take care of all six of those or uh, all five of those cases?
Peg: [pause] Uh. . . in the way that I would look at it, the way that I look at the problem, it does.
I look at the outer term and the inner term which would, the outer term that I’m looking at
is being either how you solve strictly for the power rule or for the trigonometric functions
and then going inside and actually you know. . .
I: OK. [pause] You did a very interesting thing when you solved 19.
Peg: I don’t think that’s correct.
I: I don’t think that it’s very far wrong actually. But what you did is rather than use the chain
rule, you expanded the expression. The expression was cosine cubed of t. And so you
wrote down three products of cosine t which is cosine cubed.
Peg: Uh-huh.
I: And then, you took a derivative from there.
Peg: Even with that, I didn’t finish it did I?
I: No, you needed to do the other product.
Peg: Yeah, but, yeah, I made it to that.
I: But, my question is, if you can recall, this was a long time ago, um, was there something
that didn’t say chain rule to you in that problem?
7.4 Examples of Development of a Schema 117
Peg: That would be the one that I’m still the least clear on still today because it’s not as straight
as the chain rule because the uh, the variable t doesn’t have anything associated with it like
all of the other ones, that there’s something inside the parenthesis to differentiate. Where
this one. . . so would it be 3 cosine squared t uh times minus sin? Is that the correct answer?
I: Uh-huh.
Peg: Oh.
I: That’s exactly it.
Peg: But I, it’s just not as obvious of the chain rule as all of the others. . .. (Cottrill 1999, p. 46)
In this example, Peg recognizes the chain rule in different problems. She thinks
of the chain rule as a Process she can apply to problems where she is able to
distinguish the inner and the outer functions in a composition. However, when she
is not able to make this distinction, she does not recognize their relation to the
general rule.
7.4.3 The Trans-stage
As a student reflects upon coordinations and relations developed in the Inter-stage,
new structures arise. Through syntheses of those relations, the student becomes
aware of the transformations involved in the Schema and constructs an underlying
structure. This leads to development of the Schema at the Trans-stage. A critical
aspect of the Trans-stage is development of coherence. Coherence is demonstrated
by an individual’s ability to recognize the relationships that are included in the
Schema and, when facing a problem situation, to determine whether the problem
situation fits within the scope of the Schema. In some cases, the constructions
involved in the mathematical definitions of a concept show coherence of the
Schema; this means the individual is able to reflect on the explicit structure of
the Schema and select from it the content that is suitable in solution of the problem.
The examples that follow illustrate this.
Functions At the Trans-function stage, an individual can construct various
systems of transformations of functions such as rings of functions and infinite
dimensional vector spaces of functions. The coherence of the function Schema
consists of the recognition that any function has a domain set, a range set and a
process that transforms objects in the domain set to objects in the range set
(Dubinsky and McDonald 2001).
Derivative At the Trans-derivative stage, the student synthesizes problems involv-
ing variation. For example, a student can relate the derivative as the slope of a
tangent line at a given point, with the rate of change of a function at a given point.
The student can also construct transformations among different representations of
the derivative. An individual demonstrates coherence by determining conditions
for differentiability in terms of the constructions involved in its definition (Baker
et al. 2000).
118 7 Schemas, Their Development and Interaction
Chain Rule At the Trans-chain rule stage, the student can relate function compo-
sition to differentiation and recognize that various instantiations of the chain rule
follow from the same general rule through function composition. The components
of the Schema progress from being described as a list of “inner-outer” algorithms to
a single rule, ðf � gÞ0ðxÞ ¼ f 0ðgðxÞÞg0ðxÞ, that can be applied to different situations.
The ability to grasp this general principle indicates coherence (Clark et al. 1997;
Cottrill 1999).
In the following excerpt (Cottrill 1999, p. 49), Jack gives evidence of having
constructed the chain rule Schema at a Trans-stage. This is exemplified by his
ability to group all the differentiation problems in order of difficulty using the chain
rule as a criterion and to distinguish among different instances of the chain rule such
as the power rule, the exponential rule, and rules for trigonometric and implicit
functions. He also describes the chain rule in terms of functions and their
compositions instead of making reference to external features such as parentheses.
In this excerpt, Jack uses the Leibnitz rule in an integral problem. Although he
needs some help, he is able to work on the original problem and to generalize the
chain rule to arrive at the solution.
Jack: 10, I would probably put, I don’t know, I would be tempted to put 10 in a group by itself
just because with it just being e it’s basically the chain rule, and the chain rule and the
power rule together. That’s it. That one is done. Um, 7 and, let’s see 7 requires it,
7 requires product rule and chain rule, and the all-powerful power rule. . . 8 just requires
product rule. I know, no I probably would just go ahead and group 4, 5, 8, & 9 all
together because they have the trig function and because I mean, the trig functions are
the only derivatives that can throw you off real easy if you don’t know them, because
chain rule, if you understand chain rule and you understand the use of things like
exponentials and logarithms then you aren’t gonna get messed up bad on chain rule.
You aren’t gonna get messed up bad on the power rule if you know simple mathematics.
You’re not gonna get messed up too bad on product rule as long as you remember to
keep everything straight. But, the trig functions, you know, you got, I can’t even think of
them at the moment. I haven’t used them in a while, but it’s like you know one it’s the
other and it’s just negative and one it’s just the other, period. And it’s like if you forget
that sine, the derivative of sine just by whether or not it’s got a sign change in it, you just
messed yourself up big time and you’re gonna get a wrong answer because they cycle;
and if you start off on, off with the wrong derivative of it then you have messed up the
cycle already. Then, no matter, if you know all the others you are gonna be messed up
anyway. So. . .
Jack: Um, huh, [pause] let’s see how would I write that down? OK. Let’s see. . .[pause—writing] Um, that’s just the way that it runs through in my head. [had written
f ðgðxÞÞ0 ¼ f 0 ðgðxÞÞ � ðg0ðxÞÞ � ðgðxÞÞ � x0] Without any words.
I: OK.
Jack: That’s just the way that I think of it.
I: OK. Read that for the tape.
Jack: Um, you take the derivative, essentially, whenever you use the chain rule you are
essentially looking at a function that has got another function within it um, I don’t
know, it’s sort of like doing a, handling a composite, um and in order to take the
derivative of that composite you have to first take the derivative of the outside function
and not even do anything with what’s the inside of it, the function that’s on the inside of it,
you take the outside, it’s derivative first and leave the inside function alone and then
multiply that by the derivative of the inside function, and then multiply it by the derivative
7.4 Examples of Development of a Schema 119
of the variable, or however many times you have to break it down. Because you can have a
huge function that’s got a lot of stuff inside of it and you’d have to do the chain rule
several times to get the x variable.I: OK.
Jack: So, I mean you could have, that’s just like a simple composite f of g, but you could
have, if you have like hðf ðgÞÞ then you’d have to do the derivative h with f of g inside ofit and then the derivative of f with g inside it and then the derivative of g and then
derivative of x.I: OK.
. . .
I: Can you work on this problem? [Jack was given a Leibniz rule problem with a monomial
integrand.]
Jack: This, yes. I can integrate this functions [wrote the integral]
I: Can you label the function HðxÞ?Jack: What do you mean?. . . this?
I: Yes
Jack: So this isH0ðxÞ (writesH0ðxÞ ¼ 4sin2ðxÞ cosðxÞ. So I can integrate. . . and this isHðxÞ. AndI think this is the solution.
Jack applies the chain rule in various problems, including implicit functions and
derivatives of integrals. In the excerpt, he provides evidence of having constructed
all the elements of the chain rule definition.
Table 7.1 summarizes each of the triad stages for each of the examples discussed
above.
At each stage of the triad, the student reorganizes knowledge acquired during the
preceding stage. The change from one stage to the next includes not only an
increase in the elements of the Schema but the construction of new forms of
relations or transformations among the elements of the Schema. As is evident
from the chain rule, students at the Intra-chain rule stage are able to find derivatives
of composed functions by following specific rules. The change from the Intra-stage
to the Inter-stage involves a shift of thinking that includes not only adding new
instances of the chain rule but also constructing a transformation which enables the
subject to see commonality among those rules, that is, to see those rules as specific
cases of a more general phenomenon. The change from the Inter-stage to the Trans-
stage involves the recognition of a single rule called the chain rule that applies to
any differentiable composition of functions.
The triad provides researchers with a tool with which to analyze students’
thinking and to see how it develops, taking into account the richness of problem
situations by focusing attention on relationships among different mental
constructions. It utilizes complexities involved in problem solving, how new
relationships among ideas emerge, which relationships play an important role in
newly formed structures, and development of coherence of the Schema. All of these
important aspects of a Schema are shown through students’ work in different
related problem-solving situations.
120 7 Schemas, Their Development and Interaction
Table
7.1
Comparisonoftriadstages
fordifferentSchem
as
Concept
Intra-
Inter-
Trans-
Function
Tendency
tofocusonasingle
functionand
thevariousactivitiesthatcanbeperform
ed
withit.
Possibilityto
findanalogiesam
ongarithmetic
operationsas
Processes
ondifferenttypes
offunctionsordifferencesinvolved
in
composingthem
asProcesses.Construction
ofarelationam
ongalloftheseindividual
types
offunctionsas
instancesofthesame
sortofactivity,as
ameansofconstructing
new
functionsfrom
knownones.
Constructionofvarioussystem
sof
transform
ationsoffunctionssuch
asrings
offunctionsandinfinitedim
ensionalvector
spaces
offunctions.Thecoherence
ofthe
Schem
aisdem
onstratedbytherecognition
that
anyfunctionhas
adomainset,arange
set,andaprocess
thattransform
sobjectsin
thedomainsetto
objectsin
therangeset.
Derivative
Interpretationofthederivativeas
theslope
ofthetangentlineat
specificpointsand
toperform
ActionsorProcesses
tofindthe
derivativeofafunctiononspecificintervals
todetermineifthefunctionincreasesor
decreases
onthose
intervals.Abilityto
solvesometypes
ofrate-of-change
problems.Thefocusisonindividual
ActionsorProcesses,so
thedescriptionof
thebehaviorofacurveandthesolutionof
rate-of-changeproblemsareonly
relatedby
theneedto
findthederivativeofafunction.
Abilityto
relatetheProcessofthederivativeas
theslopeofthetangentlineandtheProcess
ofthederivativeas
therate
ofchangeat
a
given
point.Constructionofarelation
betweenthem
inorder
toconsider
the
derivativeas
ameansto
describelocal
variationofthefunction.
Constructionofasynthesiswhereallthe
problemsin
whichvariationisinvolved,
such
asslopes
oftangentlines
toafunction
atagiven
pointorratesofchangeofa
functionat
agiven
point,arerelatedto
the
derivative.Abilityto
construct
transform
ationsam
ongdifferent
representationsofthederivative.Coher-
ence
isdem
onstratedbytheabilityto
determineconditionsfordifferentiabilityin
term
softheconstructionsinvolved
inthe
definitionofderivative.
Chainrule
Thestudenthas
acollectionofrulesto
calcu-
late
someindividual
caseswherethechain
rule
isusedim
plicitly,such
asthepower
ruleorthegeneralform
ula,butdoes
notsee
therulesas
beingrelated.
Recognitionthat
differentinstancesofthe
chainrule
such
asthepower
rule
orthe
derivativeofacompositionoffunctions
representsomethingmore
general.Aware-
nessthat
specialinstancesofthechainrule
arerelatedbyageneral
rule
wherethe
“outer”
partofthecomposedfunctionis
derived
andthen
multiplied
bythederiva-
tiveofthe“inner”partofthecomposition.
Abilityto
relate
functioncompositionto
dif-
ferentiationandto
recognizethat
various
instantiationsofthechainrule
follow
from
thesamegeneral
rule
throughfunction
composition.C
oherence
isdem
onstratedby
theabilityto
describetheelem
entsin
the
schem
abyasingle
rule,ðf
�gÞ0 ð
xÞ¼
f0ðgð
xÞÞg0ðxÞ
,that
canbeapplied
todifferent
situations.
7.4 Examples of Development of a Schema 121
7.5 Assimilation of New Constructions into a Schema
Construction of knowledge is a dynamic process. As individuals face new situations,
previous knowledge can be reconstructed and new knowledge can be constructed.
The notion of a Schema helps researchers understand the dynamism associated
with these changes by means of different mechanisms: as new Actions, Processes,
and Objects related to amathematical concept or topic are constructed, new relations
with previously constructed concepts are also established. New Actions, Processes,
Objects, or Schemas can become part of a previously constructed Schema, or
assimilated by a Schema that has thereby been reconstructed.
For example, an individual’s function Schema may include the definition of a
single variable function in terms of domain, range, and an idea of how domain
elements are assigned to range elements. When studying multivariable functions,
the student may assimilate these new functions as Processes into her or his previ-
ously constructed function Schema. At the same time, accommodation and
re-equilibration are likely to occur as the notion of domain is extended by
coordinating intervals of real numbers with regions in Rn. The study of linear
transformations may lead to further reconstruction of an individual’s function
Schema as the conception of both domain and range is expanded to include vector
spaces as domain and range sets. In this act of accommodation and re-equilibration,
the student learns to differentiate among different types of functions and to inte-
grate new kinds of functions into her or his Schema structure. As a result, her or his
knowledge will grow.
In the case of the derivative Schema, work with partial derivatives may not only
foster the construction of a new Schema to deal with derivatives of multivariable
functions but may also lead to the development of relations between different types
of functions and their derivatives. This assimilation results in an expanded Schema
that can be brought to bear on a wider range of problem-solving situations that
involve both single and multivariable functions.
7.6 Interaction of Schemas
In the process of learning, as knowledge develops, an individual may construct
coexisting Schemas that are constantly changing and at varying stages of develop-
ment. Each Schema is itself made up of Actions, Processes, Objects, and other
Schemas and the relationships among those structures. When facing a problem-
solving situation, a person may need to coordinate different Schemas. One goal of
research is to identify the different Schemas that need to be developed and how they
are coordinated or how they interact. Therefore, in understanding the development
of a Schema, research must not only determine how the Schema is constructed but
how it may be coordinated with other related Schemas. An example of this
relationship is considered in Sects. 7.6.1 and 7.6.2.
122 7 Schemas, Their Development and Interaction
7.6.1 Two Studies of Students’ Calculus Graphing Schema
Baker et al. (2000) and Cooley et al. (2007) described students’ attempts to solve a
non-routine calculus graphing problem in terms of the interaction of two Schemas.
In the first study, Baker et al. (2000) investigated how students coordinated infor-
mation regarding the first and second derivatives, continuity, and limits to sketch
the graph of a function. The following problem was given to the students during an
interview:
(a) Sketch the graph of a function that satisfies the following conditions:
h is continuous;
hð0Þ ¼ 2; h0ð�2Þ ¼ h0ð3Þ ¼ 0; and limx!0
h0ðxÞ ¼ 1;
h0ðxÞ > 0when� 4 < �2 andwhen� 2 < 3;
h0ðxÞ < 0 when x < �4 andwhen x > 3;
h00ðxÞ < 0when x < �4; when� 4 < x < �2; andwhen 0 < x < 5;
h00ðxÞ > 0when� 2 < x < 0 andwhen x > 5;
limx!�1 hðxÞ ¼ 1 and lim
x!1 hðxÞ ¼ �2:
(b) Do there exist other graphs besides the one you drew that satisfy the same
conditions? Justify your response.
(c) If we remove the continuity condition, and the other conditions remain, does the
graph change? In what way? Do other possible graphs exist? If other graphs
exist, could you sketch one example?
In their attempts to deal with the problem-solving situation, students had several
difficulties: they tended to work on each of the given intervals of the domain in
isolation, they failed to connect Processes related to different properties, and they
could not coordinate Processes for the properties across different intervals. Taken
together, these difficulties kept the students from drawing a correct graph of the
function. In their analysis of the data, the authors uncovered the interaction of two
Schemas as the source of the difficulties. They called these Schemas the interval
Schema and the property Schema. The authors developed genetic decompositions
for the Schemas and their interaction, which resulted in a new Schema they referred
to as the Calculus Graphing Schema.
In the second study, Cooley et al. (2007) used the same genetic decompositions
to determine whether successful calculus students’ made the same mental
constructions while working with a series of problems that included the former
7.6 Interaction of Schemas 123
one with new problems added. This offered the opportunity to examine the way
students used their knowledge when they encountered problems posed in different
representational contexts and to analyze their ability to access and use the different
parts of their Calculus Graphing Schema. The authors were interested in knowing
how the students would apply and/or reconstruct and coordinate the interval and
property Schemas when the original problem was modified. The students were
presented with tasks that increased in difficulty and that differed from the types of
tasks generally asked in calculus courses. The purpose of assigning these tasks was
to see how students would deal with the mental structures they had constructed
when encountering new problem-solving situations.
7.6.2 The Development of the Calculus Graphing Schema
The researchers found that differences in students’ difficulties and performance
could be attributed to the students’ abilities to coordinate the property and interval
Schemas. The genetic decomposition for each of the Schemas follows:
7.6.2.1 Development of the Property Schema
Intra-Property Stage: Focus on Actions or Processes corresponding to one prop-
erty of the function in isolation from other properties. Recognition that there are
other properties, but the Processes involved in them are not coordinated into a
single Process in terms of the graph.
Inter-Property Stage: Construction of relationships among some of the Processes
related to the properties of the function and transformation of these Processes into
those corresponding to the graph of the function.
Trans-Property Stage: Awareness of the transformations involved in the coordi-
nation of all the Processes related to the analytic conditions with the graphical
properties of the function in an interval. Coherence of the Schema is demonstrated
by recognition of the aspects of the graph that may be included and the coordination
of all the properties that lead to a correct graph of the function on the given interval.
7.6.2.2 Development of the Interval Schema
Intra-Interval Stage: Focus on properties of the function as Actions or Processes
on isolated intervals. The coordination of the Process or Processes related to
properties of the function over contiguous intervals has not been constructed.
Inter-Interval Stage: Grouping of contiguous intervals as a union of sets that is
part of the domain of the function.
124 7 Schemas, Their Development and Interaction
Trans-Interval Stage: Relation of all the intervals through intersections and unions
to form the entire domain of the function. Coherence of the Schema is demonstrated
by the ability to describe which behaviors of the graph are allowed by the overlap
and connection of the intervals and which are not.
When these two triads are combined in a double triad, it is possible to analyze the
interaction of both Schemas and to describe a single Schema that can be called the
Calculus Graphing Schema. Its development can be described as follows:
At the Intra-property, Intra-interval stage, the focus is on one or a few isolated
Actions on the given properties of the function on isolated intervals. The Actions
result from the relation of a single property to isolated intervals of the graph.
At the Intra-property, Inter-interval stage, the focus is on one or a few isolated
Actions on the given properties of the function on contiguous intervals seen as a
union of sets that are part of the domain of the function.
At the Intra-property, Trans-interval stage, the focus is on one or a few isolated
Actions on the given properties of the function across the domain of the function.
At the Inter-property, Intra-interval stage, some Processes related to the
properties of functions have been constructed, but the focus is on one or a few
isolated Actions of the given properties of the function on isolated intervals.
The Actions result from the relation of a single property to isolated intervals of
the graph.
At the Inter-property, Inter-interval stage, some Processes related to the
properties of functions have been constructed on contiguous intervals of the domain
of the function that are seen as being related.
At the Inter-property, Trans-interval stage, some Processes related to the
properties of functions have been constructed across the domain of the function.
At the Trans-property, Intra-interval stage, there is awareness of the
transformations involved in the coordination of all the Processes related to the
analytic conditions of the function with the graphical properties of the function in
an interval, but the focus is on one or a few isolated Actions of the given properties
of the function on isolated intervals. The Actions result from the relation of a single
property to isolated intervals of the graph.
At the Trans-property, Inter-interval stage, there is awareness of the
transformations involved in the coordination of all the Processes related of the
function with the analytic conditions of the function with the graphical properties of
the function in an interval and relations have been constructed between contiguous
intervals of the domain of the function.
At the Trans-property, Trans-interval stage, there is awareness of the
transformations involved in the coordination of all the Processes related to the
analytic conditions to the graphical properties of the function in an interval across
the entire domain of the function.
The first study (Baker et al. 2000) revealed a wide variety of differences among
students, and the genetic decomposition proved to be a useful tool to describe in
detail students’ constructions. In the second study (Cooley et al. 2007), the useful-
ness of the genetic decomposition was again proved, and more students showed
evidence of effectively describing relationships and succinctly explaining their
7.6 Interaction of Schemas 125
reasoning, showing that they had constructed a graphing Schema at a Trans-
property, Trans-interval stage. In both studies, examples of students at each of the
stages of the Calculus Graphing Schema, except for the Trans-property, Intra-
interval stage, were found. The following excerpts illustrate some of the students’
responses.
One of the students, Carol, graphed the function mainly by using information
from the first derivative. Near the graph, she constructed a table with the meaning of
the signs of the second derivative but did not use it. She described her graph as
follows:
Carol: Okay. . .it’s increasing from negative 4 to 3. . .it doesn’t say what it is past 3. Let’s
see. . .it’s increasing because it is concave up. So, that’s a point of inflection, and then
past this point it’s all down. And there is a local min at negative 2 because it’s a limit.
When trying to coordinate two conditions on a single interval, Carol said, “I don’t
understand how this [the graph to the right of x ¼ 3] can be decreasing when this [the graph
to the right of x ¼ 5] is concave up.”
Even while discussing concavity and inflection points, she did not relate these properties
to the graph and could not coordinate them. Her understanding of the first-derivative also
allowed her to integrate the inflection point at x ¼ �2 and the vertical tangent into the
graph. . .
I: Okay, now limit as x goes to 0 is infinity. Does that satisfy . . . ?Carol: Well, that means the slope would go to infinity.. . .
She then sketched a vertical line segment along the y-axis, demonstrating that she was
using the calculus graphing Schema at the Intra-property, Inter-interval stage. (Baker et al.
2000, p. 579)
John gave evidence of operating at the Inter-property, Inter-interval stage of
the Calculus Graphing Schema since he showed difficulties in coordinating
the properties on some of the intervals as well as difficulties in coordinating the
properties across the intervals:
John: Okay . . . From negative 4 to negative 2 the slope’s going to be positive, too, so this
[the graph] will go like that [in a positive slope direction]. And from negative 2 to
negative 3. . . see, that’s where I was getting mixed up because if as it [the limit of h0ðxÞapproaching infinity as x approaches 0] goes to 0, the slope’s going to infinity.
I: Right.
John: But what happens on this [the right] side of 0?
I: Right.
John: So, unless we just forget about that, you know it’s [the graph] just gotta keep increasing
. . . and so forth, up to 3. And then it just kind of turns at 3. . .up to 5, because that’s whenit’ll switch again, because of concavity here [at x ¼ 5].
I: Okay.
John: Because, wait, it can’t. All right, um, Okay. I’ve got a question for you, all right?
I: Hold on .. . .John: All right? And then according to this [h00ðxÞ changes from negative to positive at x ¼ 5],
it’s [the graph] going to switch the other way, all right? But the limit as it approaches
infinity is negative 2.
I: Right.
John: And then at 5 it switches and then it just kinda approaches negative 2 like that.
Something like that [constructs a concave-up graph dipping below the horizontal
asymptote at y ¼ �2].
126 7 Schemas, Their Development and Interaction
I: So now, does it dip down below negative 2 then?
John: I think it has to, because if it’s concave up when x is greater than 5, it’s gotta keep rising,
at least a little bit.
I: Does it?
John: Because if it was, well not to approach, to approach negative 2, it would have to go like
this [sketches a horizontal line just above y ¼ �2] and this would be a straight line.
I: Could it be concave up and still decreasing?
John: Probably. I’m sure it probably could.
I: Okay.
John: I just can’t think of how it would be.
John connected the conditions at x ¼ �4 with the graph, but he could not think of the
graph as having a cusp. He drew a smooth graph that otherwise had the required properties.
(Baker et al. 2000, p. 281)
Another student, Stacey, used mainly the first derivative, although she wrote
notes about the concavity of the graph and used those notes as she sketched the
graph. Although she worked with the union of the intervals across the domain, she
needed considerable help to coordinate the limit condition and encountered diffi-
culty in considering the conditions at x ¼ �4 (Baker et al. 2000, p. 583):
Stacy: So, how do we get [the graph] from coming in decreasing to going increasing [at x ¼ �4]
without a horizontal tangent? And it is continuous, so it can’t do one of these things . . .that point thing [cusp].
I: Well, would that not be continuous if you had a point like that?
Stacy: Okay. So it’s . . . I don’t know what I was thinking. [The graph is] smooth, that’s what
I was thinking.
Although she was able to coordinate most of the properties, she was not able to
coordinate the information at x ¼ 0 and x ¼ �4. She was also unable to transfer
information to the graph despite coordinating properties verbally. As a result, she
was deemed to be at the Inter-property, Trans-interval stage of Schema develop-
ment. This is illustrated in Fig. 7.1.
This example shows how two Schemas interact in the construction of knowledge
and demonstrates how the genetic decomposition of a Schema can be a very flexible
tool to support the study of the specificities involved in the learning of concepts.
This tool can also help in the design of teaching sequences that may help students
overcome difficulties and develop coherent knowledge. The scope of a subject’s
Fig. 7.1 Stacy’s sketch of the graph of the function (Baker et al. 2000, p. 605)
7.6 Interaction of Schemas 127
mathematical knowledge can be related to the development and interaction of
different Schemas and her or his ability to construct new relations among different
mathematical structures. The construction of Objects from Schemas is another
important feature of mathematical understanding and is described in the next section.
7.7 Thematization of a Schema
Piaget spoke about thematization in several of his books. For example, he discussed
thematization when talking about reflective abstraction and how “actions and
operations become thematized objects of thought” (Piaget 1975/1985, p. 49).
In his work with Garcıa (1983–1989), he introduced the notion of thematization
of a schema:
Abstract mathematical notions have in many cases first been used in an instrumental way,
without giving rise to any reflection concerning their general significance or even any
conscious awareness of the fact that they were being used. Such consciousness comes about
only after a process that may be more or less long, at the end of which the particular notion
used becomes an object of reflection, which then constitutes itself as a fundamental
concept. This change from usage or implicit application to consequent use, and conceptu-
alization constitutes what has come to be known under the term thematization. (p. 105)
According to Piaget and Garcıa, the development of a schema is a slow process
in which the individual becomes aware of its components and their relations. For
some time, the individual can use a schema to solve some problems without the
need to reflect on its components and the relations among them. Eventually, the
individual is able to reflect on the meaning of the components and relations that
make up the schema and is able to perform conscious actions on it. When this
happens, Piaget and Garcıa consider the schema to have been thematized. In this
sense, thematization is the mechanism by which a schema is consciously used in the
solution of problems (e.g., in Piaget and Garcıa, 1983/1989, pp. 65, 113).
In APOS Theory, thematization is associated with Schema development and its
meaning in a somewhat different way than the description given by Piaget and
Garcıa. In early APOS papers, thematization was described as one of the six kinds
of reflective abstraction (Asiala et al. 1996; see Chap. 3). In Clark et al. (1997), it
was described in the following way: “We consider a schema to have been
thematized if the individual can think of it as a total entity and perform actions
on it” (p. 353). Later thematization was seen as the mechanism responsible for
transforming a Schema into an Object (Czarnocha et al. 1999; Asiala et al. 1997a).
This same idea is expressed in Sect. 2.2. Finally, in trying to clarify the APOS
meaning of thematization, some specific research projects have asserted that
thematization occurs when Actions can be performed on a Schema, such as
operating with it, comparing it with another Schema or, as it has been described
before (Clark et al. 1997), when it can be decomposed to recover its components
(de-thematized), and/or to make the necessary Actions and Processes to reconstruct
it when conditions of the problem situation are changed. In this latter case,
128 7 Schemas, Their Development and Interaction
reconstruction involves comparison of Schemas with the same components but
different relations among them (Cooley et al. 2007). While not all these
descriptions are the same, it can generally be said that thematization is the mecha-
nism by which a Schema is transformed into an Object so that it is possible to
perform Actions on it or to apply Processes to it.
There is only one APOS Theory study focusing on thematization of a Schema
(Cooley et al. 2007). In that study, college students who succeeded in their study of
calculus were interviewed to determine whether they had thematized their Calculus
Graphing Schema. The researchers examined how students constructed relations
among properties of functions such as first and second derivatives, limits, and
continuity and how they related these properties to the graphs of functions. Nine
problems were used in the interview. The analysis of the data focused on students’
coordination of the different properties and intervals to describe possible mental
groupings within their Schemas and to determine their ability to access parts of the
Schema when called upon. In the last problem, shown below, students had to
reconsider the calculus graphing problem described in Sect. 7.6.2 when various
conditions were changed:
Problem 9:
(a) Sketch the graph of a function that satisfies the following conditions:
h is continuous;
hð0Þ ¼ 2; h0ð�2Þ ¼ h0ð3Þ ¼ 0; and limx!0
h0ðxÞ ¼ 1;
h0ðxÞ > 0when� 4 < x < �2 andwhen� 2 < x < 0
andwhen 0 < x < 3;
h0ðxÞ < 0when x < �4 andwhen x > 3;
h00ðxÞ < 0when x < �4; when� 4 < x < �2; andwhen 0 < x < 5;
h00ðxÞ > 0when� 2 < x < 0 andwhen x > 5;
limx!�1 hðxÞ ¼ 1 and lim
x!1 hðxÞ ¼ �2
(b) Do there exist other graphs besides the one you drew that satisfy the same
conditions? Justify your response.
(c) If we remove the continuity condition, and the other conditions remain, does the
graph change? In what way? Do other possible graphs exist? If other graphs
exist, could you sketch one example?
7.7 Thematization of a Schema 129
(d) If we remove all of the first derivative conditions, and the other conditions
remain, does the graph change? In what way? Do other possible graphs exist?
If other graphs exist, could you sketch one example?
(e) If we remove all of the second derivative conditions, and the other conditions
remain, does the graph change? In what way? Do other possible graphs exist?
If other graphs exist, could you sketch an example?
The solution of this problem involves the coordination of several properties that
change in overlapping intervals. The required coordinations, shown in schematic
form, appear in Fig. 7.2.
The figure delineates the interaction and overlap of the various properties across
the intervals of the domain needed to sketch the graph.
For students who were successful in solving the problem, the researchers
determined whether they accessed the necessary parts of the Schema in a flexible
way, and adapted to the demands of the specific problem situation. Once the
researchers determined that some of the students could be classified as operating
at what they called the Trans-property, Trans-interval stage of the Calculus
Graphing Schema, they considered whether thematization of the Schema had
occurred. They proposed that those students who had thematized the Schema
should be sufficiently conscious of the structure of the Schema that they could
reflect and act upon it while solving the given problem. To determine if this was
indeed the case, the researchers asked the students to reconsider the solution to all
the parts included in Problem 9. In their analysis, the researchers focused their
attention on students’ abilities to determine which properties of the graph would
change and which would remain invariant.
In general, thematization of a Schema is indicated by an individual’s awareness
of the global behavior of problems related to the Schema, flexible use of it in
Fig. 7.2 Demonstration of the coordinations needed in the solution of the problem
130 7 Schemas, Their Development and Interaction
different situations, and the ability to perform conscious Actions on it. In the study
considered in this section (Cooley et al. 2007), the researchers considered that
thematization had occurred among those students who could demonstrate aware-
ness of the global behavior of a function over its domain in terms of all the
properties across all of its intervals, that is, among students who showed that they
had constructed a coherent graphing Schema, as described in Sect. 7.6. They also
considered that thematization had occurred if a student could decompose it into its
components and analyze the relations among them to discern which of them were
relevant to the solution of the problem and to reconstruct the Schema to be used as a
totality2 for the required purpose.
As the properties of the function differed from the problem posed during the
interview, it was agreed that a student who had thematized the Schema should give
evidence of having coordinated all given properties across all intervals of the
function. According to the authors, the ability to make this coordination in lieu of
changes made to the properties of the function demonstrated conservation in their
understanding. For the students who demonstrated such evidence, the authors
asserted that the Calculus Graphing Schema had become a fundamental part of
the students’ understanding and could be viewed as an Object, that is, the Schema
had been thematized.
In this study, the focus was on finding students who had thematized the Schema
and not necessarily whether they viewed it as a totality. More research is needed to
understand the differences between these two concepts. In the next set of excerpts,
Susan shows a Trans-interval, Trans-property level of Schema development but
fails to achieve thematization. In the following passage, Susan gives evidence of
coordinating all the given conditions across the intervals and of synthesizing the
different transformations required to graph the function. However, when asked
about the function’s behavior if the continuity property is removed, she has
difficulty considering which properties of the graph would remain invariant and
which would change. Although she is conscious of her Calculus Graphing Schema,
her inconsistency in performing Actions on the Schema suggested that
thematization has not yet occurred.
We can observe this difficulty when Susan worked on the following problem
(Cooley et al. 2007, p. 10).
Problem 7c.
Sketch the graph of a continuous function that complies with the following
conditions:
its domain is� 1 to 1;
2 The term totality was used as part of the encapsulation of a Process into an Object before it was
proposed as a new stage in Dubinsky et al. (2013). The former is the meaning of its use here. In
Chap. 8, this term is used differently as a possible new stage in APOS Theory between Process and
Object.
7.7 Thematization of a Schema 131
it is increasing on ð�1; 0Þ; decreasing on ð0; 1Þ;
concave down on �1;� 1
2
� �and on ð0; 1Þ; and concave up on � 1
2; 0
� �
What happens at the point x ¼ � 12of the function?
What happens at x ¼ 0?
Is this function unique? How could you change the graph if you were allowed to
remove the continuity condition? Justify your response.
If other graphs exist, could you sketch an example?
I: And the continuity condition? If you remove it, what happens?
Susan: I remove it and. . . well. . . in order for it to be discontinuous I can take off the definition
on this interval, the one at the middle. So it has a jump and it’s not continuous and it’s
defined on the other two intervals of the domain.
I: Well, I think that the question is not clear. The idea is that everything remains the same, I
mean the domain where it is defined, the function, and also its properties, where it
increases or decreases, and so on. The only thing that can change is that it does not have
to be necessarily continuous like the one you drew here.
Susan: Oh! Now I understand, then the other thing I said is also wrong. That is, the function is
like this one, that I drew here and it’s one function, not three. Of course, it’s defined by
parts and what happens, let’s see, well it’s still not unique, but it is so because we don’t
know the values it takes at each point. We only know that it grows, or decreases, and
so on. Then the graph could be up here or down here and there could be a lot of graphs, a
lot of functions that satisfy those conditions. And if you remove the continuity condition,
well, I don’t know. Let’s see, I think that at � 12and at 1, at those points there are those
sharp points and we don’t know what happens, we only know what happens in the open
intervals. Yes, I think there can be little holes there, aren’t there? The rest. . . but. . . itsdomain is from � 1 to 1. That means it is defined and if there is a hole it would not be
defined. Oh well, it would be a little weird, but it could have a little hole in those points
and those points can be defined up here or down here so that there is not a jump. I think
so. I think that’s right. In reality I don’t know. I don’t understand very well. This is
difficult, isn’t it? Yes, well, I don’t know, I better continue and do number 8, Okay?
I: Yes, if you prefer to, but why don’t you think a little bit more on this one?
Susan: The problem is I don’t know how. Because what I do know is that at these points there is
a derivative there, but at these other points there is not and. . .I: The fact that the derivative is defined there, does that tell something to you?
Susan: Something? Yes, well about continuity, there is a theorem, but I can’t remember it. Well,
I think I better do the next one.
I: Alright.
On Problem 9, Susan constructs an accurate graph (Fig. 7.3) and describes
coordination of properties across intervals. However, she again struggles to inte-
grate the properties across the intervals when some conditions were removed.
Taking into account all her responses, the researchers concluded that even though
she demonstrated a Calculus Graphing Schema at the Trans-interval, Trans-
property level, she was unable to perform the Actions needed to break the Schema
into its components and to distinguish between the aspects of the graph that
remained invariant. As a result, the researchers decided that her Schema had not
been thematized.
132 7 Schemas, Their Development and Interaction
Only one of the 28 interviewed students, Clara, gave evidence of thematization.
She was able to describe which properties of the function remained unchanged
when changes to the problem were introduced. She explained in detail the effects on
the graph resulting from each change, that is, she showed that the Schema was an
Object to her, as can be seen in the following excerpts from her interview (Cooley
et al. 2007, p. 13):
On Problem 7 described above, she responds to the removal of continuity saying:
I: . . . Then what happens if you can remove the continuity condition?
Clara: If the continuity condition is removed at, for example, at x¼ 0, the function will still be a
function that does not have derivative at 0, like this for example [sketches a possible
graph around the point], or it may be like this other one, too [draws another possible
graph around the point], because there would not be a derivative at x ¼ 12; or� 1
2. Then
we would not be able to say that it is an inflection point because the second derivative
would not exist either.
I: Is there a condition that says that the derivative does not exist at � 12?
Clara: Yes, oh no, there is no condition, but if you remove continuity you can break the curve at
those points where there is no derivative and then. . . oh I see, I am wrong there, because
at that inflection point the first derivative has to be something, for example 0, and then it
has to be continuous. It is only at 0 where there can be a change for the function if the
continuity is removed.
Clara explained which properties remained fixed and the intervals to which those
properties applied. She was able to remove any condition and flexibly reassemble
the information into a new graph and explain why the resulting graph is consistent
with the given information. This is exemplified by her work on Problem 9:
Clara: No it’s not unique (referring to the graph of the function), but all the graphs have to have
the same basic shape as this one. . .I: And what happens if we remove the continuity?
Clara: Then it can change. The interval is open at �2. We know the derivative is zero and
therefore the derivative exists. So it means that the function must be continuous there [at
x ¼ �2], and the same happens at 3 and at x ¼ 0 because of the limit condition. Because
the limit of the derivative has to be infinite, we can have a discontinuous function there
[at x ¼ 0]. It can have an asymptote there [x ¼ 0] but still h of 0 has to be 2 and the
function has to grow on both sides of 0. So the conditions are satisfied.
Fig. 7.3 Susan’s work for Problem 9 (Cooley et al. 2007, p. 11)
7.7 Thematization of a Schema 133
I: Is x equal to 0 the only point where you can break the graph if the continuity condition isremoved?
Clara: Let me see, no I skipped �4. I don’t have any condition for the derivative there [at
x ¼ �4] and so the graph can also be broken there. There can also be an asymptote there.
(Cooley et al. 2007, p. 14).
Clara’s ability to break the problem into its parts and to introduce changes
dictated by the new conditions is illustrated in Fig. 7.4. She provided reasons for
her decisions and showed that she was able to differentiate those parts of the
Schema needed in each situation and to integrate them back into a new graph
(Cooley et al. 2007, p. 16):
I: Now if we have a continuous function and we remove the conditions on the first
derivative, what happens?
Clara: If we remove the conditions on the first derivative. Let me see, we still have h(0) is 2 andthe conditions on the second derivative are the same, the conditions on h double prime
mean that at x ¼ �4 the derivative is not defined, but without the conditions on h prime
the function can be decreasing in the interval from �4 to �2, and also we don’t have to
have the maximum at 3 and we can have a graph like this one.
I: Okay, now the last part, imagine now that the conditions on the second derivative are the
ones that are removed. What would happen to the graph of the function?
Clara: Now if we don’t have the conditions for the second derivative we still have the value for
the function at 0 and the limit of the derivative at 0 and then the function decreases from
�infinity to �4 and decreases from, no, increases from �2 to 0 and from 0 to 3 and
decreases from 3 to infinity, but approaches y¼�2 at the right end of the graph. There is
an asymptote there [at y ¼ �2], so we could have a graph that looks like this. There can
Fig. 7.4 Work illustrating
Clara’s thematized schema
(Cooley et al. 2007, p. 15)
134 7 Schemas, Their Development and Interaction
be changes in the concavity between �2 and 0 and between 0 and 3. Only the first
derivative there will not be zero and also there has to be an inflection point because of the
asymptote there at the right end of the graph but if we don’t have those conditions, the
change in concavity can be at any place to the right of 3, even at 3. Well, not at 3 exactly
because there the derivative is zero and it decreases after 3, but to the right it is possible
that there is an inflection point. Well, yes, x ¼ 5 does not have to be an inflection point
and x ¼ � 4 is not necessarily a cusp. And also, there can be other inflection points and
still satisfy the other conditions, this gives you more freedom to change the graph. For
example, at x ¼ 5 we don’t have any restriction now.
Clara demonstrated that the relations among the concepts in her Schema were
stable and that the different parts of her Schema could be accessed and reevaluated
appropriately. She was able to act on the Schema as an Object. In her explanations,
she demonstrated conscious control of the result of those Actions. The researchers
considered this to be clear evidence of thematization.
The fact that it was possible to find a student who thematized the Calculus
Graphing Schema provides evidence that thematization of a Schema is possible. It
also shows that it is possible to find evidence of students’ conscious and flexible use
of mathematical knowledge, although more research is needed to see how prevalent
this level of understanding is.
The only student in this study who showed evidence of thematization of the
Graphing Calculus Schema had already taken three calculus courses and an analysis
course. This fact may demonstrate that the thematization of a Schema takes time
and that many opportunities for reflection as well as good instructional strategies
are needed to accomplish it. More research is needed, however, to back this
conclusion.
Appendix: Problems for the Interview in the Chain Rule
Study (Cottrill 1999)
Compute the derivative of each of the following functions. Show all your work.
11. f ðxÞ ¼ 11x5 � 6x3 þ 8 12. gðxÞ ¼ 3=x2
13. hðxÞ ¼ ðx2 � 3Þ 14. y ¼ 3ex � 4 tanðxÞ15. y ¼ x2 sinðxÞ 16. FðxÞ ¼ ð1� 4x3Þ217. GðxÞ ¼ 2 5x2 þ 1ð Þ4 � 4x 5x2 þ 1ð Þ4 18. HðxÞ ¼ sin ð5x4Þ19. y ¼ cos3ðtÞ 20. y ¼ e�t2
Additional question for interview:
Compute F0ðxÞ if FðxÞ ¼ðsin x0
et2
dt
Appendix: Problems for the Interview in the Chain Rule Study (Cottrill 1999) 135
Chapter 8
Totality as a Possible New Stage and Levels
in APOS Theory
The focus of this chapter is a discussion of the emergence of a possible new stage or
structure and the use of levels in APOS Theory. The potential new stage, Totality,
would lie between Process and Object. At this point, the status of Totality and the
use of levels described in this chapter are no more than tentative because evidence
for a separate stage and/or the need for levels arose out of just two studies: fractions
(Arnon 1998) and an extended study of the infinite repeating decimal 0:�9 and its
relation to 1 (Weller et al. 2009, 2011; Dubinsky et al. 2013). It remains for future
research to determine if Totality exists as a separate stage, if levels are really
needed in these contexts, and to explore what the mental mechanism(s) for
constructing them might be. Research is also needed to determine the role of
Totality and levels for other contexts, both those involving infinite processes and
those involving finite processes. It seems clear that explicit pedagogical strategies
are needed to help most students construct each of the stages in APOS Theory and
that levels which describe the progressions from one stage to another may point to
such strategies. Moreover, observation of levels may serve to help evaluate
students’ progress in making those constructions.
Evidence for levels and Totality in the studies of the relation between 0:�9 and
1 was mentioned in Chap. 5 and is discussed in this chapter; evidence for levels
from the study of fractions is considered in Chap. 9.
This chapter contains three themes: a general discussion of the progression
between stages in APOS Theory and related pedagogical strategies; a description
of the terms stages and levels as they appear in the work of Piaget and in APOS-
based research; and a summary of the research regarding0:�9 that suggested the needfor levels between stages and for Totality as a new stage between Process and
Object. Because the idea of Totality and related notions have appeared in previous
research, both within and outside of APOS Theory, it is necessary to point out the
differences between those usages and the proposed meaning of Totality. Included in
this discussion is a review of the tentative nature of Totality as a stage and
comments on the need for future research to determine its status.
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_8,© Springer Science+Business Media New York 2014
137
8.1 Progression Between Stages
One major issue in APOS-based research is to understand the cognitive progression
from one of the stages, Action, Process or Object, to the next “higher” stage. Very
often, in learning a particular concept, a subject achieves the Action stage but shows
difficulty in reaching Process, or reaches the Process stage but cannot progress
to Object. How can instruction help students overcome these apparent obstacles? To
answer such a question, it is first necessary to understand why the difficulty occurs.
That is: How do themechanisms that lead from one stage to the next (interiorization—
from Action to Process and encapsulation—from Process to Object) function?
There are some “first-tier” answers to these questions. In the case of progression
from Action to Process, an individual may fail to develop a Process conception
because he or she has not yet successfully interiorized the Action. One powerful
pedagogical strategy to help students make this mental construction, discussed in
Chap. 5, is to have them represent the action as a computer procedurewhich accepts an
appropriate input, performs the action on it and returns the result. Research has shown
that writing these types of computer programs helps students to move from Action to
Process (Weller et al. 2003). For children at the age of concrete operations (in the sense
of Piaget, 1975), who may not yet be capable of writing such computer programs, the
teaching sequence may need to start with an Action that can be imagined. The role of
imagination in the interiorization of an Action is considered in Chap. 9.
In the case of progression fromProcess toObject, an individualmay fail to progress
to anObject conception of a Process because he or she has not successfully constructed
and applied a transformation to the Process. Again, a strategy, also discussed in
Chap. 5, exists with computer programming: once a Process has been represented as
a computer procedure, the student can write computer code to transform the computer
procedure in various ways (provided the programming language is sufficiently pow-
erful). As indicated in Chap. 5, research has shown that this type of activity facilitates
encapsulation of the Process into an Object (Weller et al. 2003).
But what about the substantial number of students who are not helped by these
instructional treatments (Weller et al. 2003)? As indicated above, the search for an
answer to this question must begin with an investigation of a previous question:
Why do difficulties in moving from Action to Process, and from Process to Object,
exist? The investigation into this latter question begins with an attempt to better
understand the kinds of thinking that may be taking place as an individual tries,
successfully or unsuccessfully, to progress from one stage to another. But before
discussing investigations of progressions between stages, determination of what a
stage is, what a level is, and the differences between them must be considered.
8.2 Stages and Levels
In his work, Piaget considered not only stages, but levels between stages. In the
APOS-based studies of pre-service teachers’ understanding of the relation between
0:�9 and 1 (Weller et al. 2009, 2011; Dubinsky et al. 2013), and of children’s
138 8 Totality as a Possible New Stage and Levels in APOS Theory
development of the concept of fraction (Arnon 1998), the data suggested the
existence of levels between stages. In this context, a stage refers to one of the
mental constructions of Action, Process, or Object, and a level denotes a develop-
mental juncture between two of these stages. In considering both levels and stages,
a full investigation of an individual’s development of her or his understanding of a
mathematical concept would include what it means to progress between stages,
between levels, from a level to a stage, and from a stage to a level. Sections 8.2.1
and 8.2.2 contain a more detailed discussion of levels and stages as they appear in
Piaget’s work and in the studies of 0:�9 and its relation to 1.
8.2.1 Piaget’s Work on Stages and Levels
The following comments are based on the work of Piaget (1974/1976, 1975) and
Dubinsky et al. (2013).
A stage cannot be skipped. If it is, the subject’s understanding of the concept will
lack coherence. Thus, stages are sequential, with each stage necessary for develop-
ment of successive stages.
A level may or may not be reflected in the data of a specific subject. This is
because the subject may be able to move to the next level or stage rapidly so that the
level is skipped, done very quickly, or is not observable in the already acquired
higher level or stage.
Stages are invariant over topics and are part of the general theory. Levels will
be different for different concepts (Dubinsky et al. 2013). In many works, Piaget
gave examples in which the development of different concepts gave rise to different
levels.
The role of the level is to analyze, and provide mechanisms for, building the next
level in a stage or the stage itself; this should be reflected in the definition of the
level. According to Piaget, stages, together with their levels, are sequential, each
contributing to the development of its successor. In particular, every level
contributes to the development of the following stage.
8.2.2 Levels in APOS Theory
The use of levels in APOS-based research in the study of fractions (Arnon 1998)
and infinite processes (Weller et al. 2009, 2011; Dubinsky et al. 2013) is quite
consistent with what Piaget wrote, as described in the above summary. In her
study of fractions, Arnon investigated the learning of several fraction concepts.
For these concepts she found, within the progression from Action to Process, what
she called “more subtle distinctions”. These constitute what she referred to as a
“continuum”, and can be considered to be levels within the progression from the
8.2 Stages and Levels 139
Action stage to the Process stage. (See Chap. 9 for more details on these studies).
On the other hand, in the studies related to 0:�9, different levels were found betweenall of the different stages.
The levels defined in APOS-based research, as discussed here and in Chap. 9, are
content specific and arose from the interview data. Hence, there is no expectation
that the levels reported in Dubinsky et al. (2013) and Arnon (1998) will necessarily
be found in studies of other topics, even those involving infinite mathematical
processes or fractions. This aligns with Piaget’s ideas above and with his other
work. Stages, as cognitive developments of knowledge of specific mathematical
concepts, are defined in terms of major structures, which are general and do not
depend on specific content. Levels, on the other hand, as indicated above, depend
on the specific topic and the data collected from several subjects. For example, in
the experiment called The Hanoi Towers (Piaget, 1974/1976), there are no
sub-divisions of stages into levels, and all the data is presented in just the three
main stages I, II, and III. In the experiment Walking on All Fours (Piaget, 1974/
1976) stages I and II are each divided into two levels, denoted IA, IB, and IIA, IIB.
In the experiment on Seriation (Piaget, 1974/1976), the data of stage I has two
layers of levels, so that within stage I there are four levels, IA(i), IA(ii), IA(iii),
IB. In both Piaget’s work and in APOS-based research, the definitions of the levels
within stages are based on interview evidence.
8.3 A New Stage in the Infinity Studies
In addition to levels between stages, the widespread difficulties of students in
progressing from Process to Object conceptions led to consideration of another
possible change in APOS Theory. Obstacles in this progression appeared in the
studies of the relationship between 0:�9 and 1 (Weller et al. 2009, 2011; Dubinsky
et al. 2013) which report on the difficulty of this progression in specific mathemati-
cal contexts and tend to confirm the results of previous studies (e.g., Sfard 1991;
Breidenbach et al. 1992). Indeed, Sfard even suggests that this progression from
Process to Object “seems inherently so difficult that at certain levels, it may remain
practically out of reach for certain students” (Sfard 1991, p. 1).
The results of infinity studies related to historical developments of the concept of
infinity (Weller et al. 2004; Dubinsky et al. 2005a, b) suggest that the difficulty of
the progression from Process to Object may be particularly strong for infinite
processes. Brown et al. (2010) acknowledged this difficulty and called the Object
in this case a transcendent object. It has the property of being very different from
any of the objects in the sequence making up the infinite process. A large percent-
age of the subjects in Dubinsky et al. (2013) reached the Process stage but not the
Object stage. Among those who did not make this progression, there were
differences in their interview responses. The data suggests that one way to interpret
these differences is to posit the existence of a new stage, Totality, between Process
140 8 Totality as a Possible New Stage and Levels in APOS Theory
and Object, and then study the two progressions from Process to Totality and from
Totality to Object. Figure 8.1 is a variation of the diagram in Fig. 2.1 that would
incorporate this new stage.
8.3.1 The Introduction of a New Stage
The literature reports no more than limited success in helping students overcome
their difficulties in progressing from a Process to an Object conception of 0:�9 .Although Zazkis and Leikin (2010) and Weller et al. (2009, 2011) are exceptions,
the progress reported in these three studies fell considerably short of a complete
solution to the problem. In particular, in the APOS-based studies by Weller et al.
(2009, 2011) of pre-service elementary and middle school teachers, some students
who completed APOS-based instructional treatments made somewhat more prog-
ress in development of an Object conception of 0:�9 and belief that 0: �9 ¼ 1 than did
students who completed traditional instruction, but many did not. Carly is one
example of the latter group. She gave substantial evidence of seeing 0:�9 as a
Process. In her interview, she repeatedly spoke about the idea that 0:�9 “keeps
going”. The following excerpt provides one such instance:
I: If I give you a decimal point and I give you one hundred 9s after that decimal point, is
that .9 repeating?
Carly: To a certain point, but it ends. This one [0:�9] is infinity. This one always keeps going.
Expressions that relate to 0:�9 going on forever were considered to be indicationsof a Process conception. On the other hand, although given ample opportunity to
Fig. 8.1 APOS Theory with Totality
8.3 A New Stage in the Infinity Studies 141
speak about0:�9as an Object, many subjects like Carly spoke exclusively in Process-
oriented language, as exemplified in the following passage:
Carly: Just because it’s 0:�9 still not one. One is a whole number. One is one and this is
approximately.
Here, Carly is not only denying the equality of 0:�9and 1, but may also be rejecting
the idea that 0:�9 is even a number. Also missing with subjects such as Carly was any
evidence of thinking of, or of constructing, transformations to act on 0:�9.The following excerpt from Tanisha is a similar example. She repeatedly stated
(here and elsewhere) that 0:�9 just keeps going on forever, but is unable to operate on
it in order to solve the equation 0:�9þ X ¼ 1 for X:
I: Yeah. Uh-huh. Now, so if you’ve got this equation—.9 repeating plus X equal 1, what
do you think goes in for X?Tanisha: Awesome. I want to say .1 repeating. Just because that will obviously make it 10, but I
don’t—I don’t know what X could be. That’s the part that I’m missing. What’s
between the 9 and what’s the— you know, the .9 and the 1, what’s in between there.
I: And what makes it so that you can’t determine that?
Tanisha: Because the 9 keeps going and I can—I mean— if you told me that if you really
wanted it to go on till 10,000, I could find what that X would be, but since it keeps
going on forever then the 9 doesn’t stop, I guess, working. And so X—not that it no,
not that X would always change, but it would always keep going with the 9.
These two examples contrast with subjects such as Estelle. Like Carly and
Tanisha, Estelle sees 0:�9 as a Process (as seen in the excerpt Estelle: Process), but
differs from Carly and Tanisha in her ability to conceive of 0:�9 as an Object and in
her expression of a belief that 0:�9 ¼ 1 (as seen in the excerpt Estelle: Object):
8.3.1.1 Estelle: Process
Estelle: Well, if you keep adding 9’s and the sequence like, never ends, but—I— that’s a hard
question.
I: That’s hard. Yeah, that’s hard.
. . .Estelle: Because you’re always gonna add one more 9.
. . .
Estelle: No. I think it’s more that since sequences go on forever, that you’re forever gonna be
adding one more 9.
. . .
Estelle: Yeah. And that’s gonna go on forever so you’re always gonna keep borrowing and
borrowing and borrowing.
8.3.1.2 Estelle: Object
Estelle: Okay. So we have .9 repeating plus X is equal to 1. So to solve for X you subtract .9
repeating from both sides and X is going to equal 0, because you have like—yeah. Yes.
. . .
Because with .9 repeating like, I’ve always been taught that like, the bar over the
9 means it goes on forever, and so if you have an infinite—if you have the bar over it,
142 8 Totality as a Possible New Stage and Levels in APOS Theory
it’s gonna be infinite so you’re gonna have all those 9’s at once, I guess. And that’s
equal to 1.
I: And how do you know it’s gonna equal 1?
Estelle: Because .9 repeating equals 1 because it goes on forever.
The infinite amount of 9’s that’s a number. Yeah.
The ability to apply operations to the Process in question, or, in the case of 0:�9,referring to it as a “number” or a “thing” were considered to be indications of an
Object conception; inability to perform such operations were considered to be
evidence to the contrary.
Estelle’s comment in the above excerpt that “it’s gonna be infinite so you’re
gonna have all those 9’s at once” seems to be a key to success in, and a better
understanding of, the progression from Process to Object conceptions. Subjects who
indicated an inability to see all of the 9’s at once failed to reach anObject conception.
Some subjects who indicated that they had this ability went on, like Estelle, to
achieve an Object conception, but some, although sharing the ability to see all the 9s
at once, did not. This suggested the possibility of a new stage between Process
and Object. This new stage, Totality, refers to the ability, as expressed by Estelle, to
see or to imagine all of the 9s present at once. Estelle appears to have achieved the
stage of Totality and also to have gone on to achieve the Object stage for 0:�9The following passage presents yet a different example. Here, Natasha gives
evidence of Totality when she indicates that she may be conceiving of all of the 9’s
at once:
Natasha: If you went on forever, at the end of forever then it [0:�9] would be the whole thing
But, when asked to determine the solution to the equation 0:�9þ X ¼ 1 she has
difficulty:
I: Ok. Your thing is, you can’t put anything else in there [for X]?Natasha: Yeah. Even though I don’t think there is a number such as that [0:�011] because that
means this would have to end. But the 9’s never end so why would the zeros have to
end? And just imagine that 1 out there, at the end of the 0s. Or, you can imagine this,
and the one being like the end and the zeros going on, like just pushing the 1 back,
because of all the 0s forever.
Although Natasha achieved the Totality stage, the second of her two excerpts
suggests an inability to see 0:�9 as an Object.
Subjects like Natasha, who moved beyond Process toward Totality but did not
reach Object, provide support for the possibility of Totality as a definite stage
between Process and Object.
1 0:�01 refers to repeating 0s, with 1 at the end.
8.3 A New Stage in the Infinity Studies 143
8.4 Levels Between Stages in 0:�9
The data in Dubinsky et al. (2013) also uncovered the existence of levels between
stages, that is, incremental points of progression from Action to Process, from
Process to Totality, and from Totality to Object. In Dubinsky et al. (2013), there is a
very detailed description, called a Framework for Analysis (FFA), that gives
operational definitions that provide an objective and reasonable set of criteria
for determining all levels and stages in the context of an infinite repeating decimal.
For most subjects, the criteria sufficed to determine the level, but in a few cases it
was necessary to return to the interviews and make interpretations. Interpretations
of the excerpts that follow provide the rationale for the operational definitions
outlined in the FFA that were used to determine the levels between stages.
8.4.1 Action to Process
There were some subjects who gave evidence of an Action conception by writing
out a finite number of 9s, but gave no evidence of a Process conception, or any stage
beyond Process. These subjects had progressed to the stage of Action but not
beyond. On the other hand there were subjects who gave substantial evidence of
having achieved an Action conception and having progressed to the Process stage.
However, there were also subjects for which the determination was less clear. For
example, Maria made multiple statements in which she expressed the idea that
the 9s in 0:�9 continue forever, a strong indication of Process. But she also stated thatshe believed that a finite number of 9s would suffice:
I: So, do you think .999 is equal to .9 repeating?
Maria: Hmhm . . . Well, maybe not .999, but definitely after maybe four or five nines.
Excerpts such as this suggested that Maria had not fully achieved a Process
conception of 0:�9 but was in transition from Action to Process. This suggested the
existence of a level between Action and Process called Emerging Process (EP).Table 8.1, based on the FFA (Dubinsky et al. 2013), summarizes the operational
definitions of the progressions between stages and/or levels in the transition from
Action to Process. In this table, “Segment” refers to a short, coherent episode in the
transcript that carries with it a very brief (most often a single sentence or phrase)
descriptor to summarize its content. The arrows indicate transitions between levels
and/or stages.
8.4.2 Process to Totality
As seen in the examples given above, among the subjects who reached the Process
stage, several gave evidence of seeing, or beginning to see, the repeating decimal
0:�9 as a totality. Others gave no such evidence. To study more closely the
144 8 Totality as a Possible New Stage and Levels in APOS Theory
progression between these two stages, three levels were introduced: Start towardsTotality (ST), Progress towards Totality (PT) and Emerging Totality (ET).Following are some examples.
Carlos made many statements about 0:�9 going on forever. For example,
Carlos: Because, again .9 is . . . we’re talking about .9 repeating, so you’re gonna just keep on
repeating and repeating that space. So it just keeps on going and going and going, so it’s
always changing.
So he was considered to have achieved the Process stage. For the next stage—
Totality—it was a different story. One of the indicators for Totality involved
analysis of the results of the following thought experiment: divide a square into
10 equal parts, shade 9 parts, divide the remaining part similarly and repeat this
process indefinitely. Subjects were asked to imagine how much of the square would
be shaded. Shading of the total square indicated Totality for the process (but not
Object because there is no application of an action or process to the process); a part
left unshaded indicated otherwise. When discussing this question, Carlos once
expressed the opinion that “all of it” would be shaded, but twice asserted that a
little bit was always left. The inconsistency of his responses, together with the
higher relative frequency of his difficulty in seeing the terms of 0:�9 “all at once,”
suggested no more than a Start towards Totality (ST).
Rosa, on the other hand, seemed to express equally often positive and negative
opinions about all of the 9s being present all at once. For example, at one point she
says:
Rosa: I can’t imagine all the 9’s that it would take to get to forever.
But when asked about the thought experiment, she asserts that the entire square
would be shaded. The relative equality of the number of positive statements (the
ability to see 0:�9 “all at once”) and negative statements (inability or difficulty in
seeing 0:�9 “all at once”) suggested Progress towards Totality (PT).
Table 8.1 Progression from Action to Process
Progression
A: Action
EP: Emerging Process
P: Process
Criteria
A # Segments under Evidence of Action > 0; and # segments
under Evidence of Process ¼ 0
A ! EP # Segments under Evidence of Process > 0 and one or more
interview segments indicating difficulty in making the transition
from A to P (e.g., the subject thought in terms of action when
a process was called for)
A ! EP ! P # Segments under Evidence of Process > 0 and no interview
segments indicating difficulty in making the transition from A to P
8.4 Levels Between Stages in 0:�9 145
Susan made statements such as:
Susan: You can’t really imagine never ending nines.
But on twice as many occasions she used phrases such as “when you actually had
.9 repeating and it’s never ending” and she asserted that the entire square would be
shaded in the thought experiment. Although her responses revealed difficulties, the
relative dominance of positive versus negative comments indicated an Emerging
Totality (ET) level.
Finally, subjects who gave indications of seeing the 9s present all at once and no
evidence to the contrary were deemed to have reached the Totality stage.
Table 8.2, also based on the FFA (Dubinsky et al. 2013), summarizes
the operational definitions of the progressions between levels and/or stages in the
transition from Process to Totality. In this table, “Segment” again refers to a short
portion in the interview transcript and the arrows indicate transitions between
stages and/or levels.
8.4.3 Totality to Object
In APOS Theory, moving from a Process conception to an Object conception is
indicated by the individual’s ability to think about and/or perform actions or
processes on the Process. Among the subjects who achieved the Process stage,
several achieved some of the levels in the progression from the Process to the
Totality stage, and then went on to show progress toward the Object stage. How-
ever, that progress was tempered by evidence of difficulties. Similar to the
examples above, which showed differing levels of progress from Process to Total-
ity, the interview evidence suggested the need to introduce intermediate levels in
Table 8.2 Progression from Process to Totality
Progression Criteria
ST: Start Toward Totality
PT: Progress Toward Totality
ET: Emerging Totality
TOT: Totality
For any level in the progression from Process to Totality
it is required that the # segments under Totality Sees > 0
(Here, “Totality Sees” means the subject sees the Process
as a Totality and “Totality Does not See” means
the subject does not see the Process as a Totality)
ST # Segments under Totality Does not See > # segments
under Totality Sees
ST ! PT # Segments under Totality Does not See ¼ # segments
under Totality Sees
ST ! PT ! ET # Segments under Totality Does not See < # segments
under Totality Sees
ST ! PT ! ET ! TOT # Segments under Totality Does not See ¼ 0
146 8 Totality as a Possible New Stage and Levels in APOS Theory
the progress to the Object stage. The following levels were indicated: Start towardsObject (SO), Progress towards Object (PO), and Emerging Object (EO).This decision was confirmed when it was seen that there were subjects whose
progression went no further than one of these levels. Following are some examples.
Awaethu was able to see that X ¼ 0 is the solution to the equation 0:�9þ X ¼ 1.
She had difficulties, however, with manipulating 0:�9 directly to show that it was
equal to 1 because she saw 0:�9 as a “repeating thing”. Therefore she was designatedas having made a start towards Object (SO).
Roberto, on the other hand, was evenly balanced in giving evidence of having
achieved an Object conception and of not having reached that stage, so he was
designated as Progress towards Object (PO).
Although Rita gave indications of an Object conception, for example by
expressing her belief that 0:�9 is equal to 1, and no indications to the contrary, the
weakness of her positive comments caused her to be one of the few subjects for
whom determination of level required interpretations of her comments beyond mere
application of the FFA. She made statements about the 9’s in 0:�9 and the 0s in 0:�0
stopping or not stopping. She also said that because 0:�9 “doesn’t stop, you’d get 1”.The last statement suggests Object; the previous one reveals confusion. For exam-
ple, Rita said:
Rita: But in the case of 1 minus .9 repeating, the 0 would just continue on forever until the 9s
stopped, which then would make those two numbers not equal to each other so you’d have
a 1 at the very end.
Rita’s tendency to see the difference between 0:�9 and 1 as 0:�0 and her inability tosay 0 indicated some difficulty with encapsulation. However, according to the FFA
criteria, her overall progress, marked by the fact that she made exclusively positive
statements, including an ability to compute with 0:�9 , suggested Object. The
apparent difficulty with encapsulation necessitated the need for interpretation,
which led to a designation of Emerging Object (EO).
Rita: You can give me an equation and I will believe the equation. You can give me the
numbers to prove that they are equal, but the way I see it, the simplest way, is 1 minus .9.
As long as the 9 doesn’t stop, the difference is going to be point zero repeating.
Finally, Clyde performed correctly at least 10 arithmetic operations on 0:�9 and
located it in an appropriate position on the number line. In the following exchange,
he symbolizes 0:�9 as an object X in an equation.
Clyde: Mentally, I think of it more as X, like in an equation.
I: Uh-huh.
. . .
Clyde: Because dealing with .9 repeating, just trying to even visualize it, it’s a little bit hard,
so I think of it as X—an X sign. That way there’s a place holder in my head that I can
think about it that’s more concrete than dealing with the .9’s repeating.
Because he gave exclusive evidence of having reached object, with no evidence
to the contrary, he was designated as having an Object conception of 0:�9.
8.4 Levels Between Stages in 0:�9 147
Table 8.3, based on the FFA (Dubinsky et al. 2013), summarizes the operational
definitions of the progressions between stages and/or levels in the transition from
Totality to Object.
8.5 Previous Uses of the Idea of Totality
Research much earlier than Dubinsky et al. (2013) discussed ideas similar to the
introduction of a stage between Process and Object. Dubinsky referred to the
notion of a function as a Totality, so that it can be a point in a function space, as
opposed to a process (Dubinsky 1984), and of an individual’s inability to see a
process as a “total entity” and reason about it (Dubinsky 1987); Ayers et al. (1988)
referred to the encapsulation of a process into a single, total entity and thinking of
it as a mental object; and Cornu and Dubinsky (1989) distinguished between
performing an action in a step-by-step manner and seeing it as a totality. All but
the last of these examples appear to include a notion of Totality as part of an
Table 8.3 Progression from Totality to Object
Progression Criteria
SO: Start Toward Object
PO: Progress Toward Object
EO: Emerging Object
O: Object
For any level in the progression from Totality to
Object it is required that #segments under Object
Sees > 0
(Here, “Object Sees” means the subject does per-
ceive the process as an object, “Object Tries to
See” means that the subject tries to see the process
as an object and “Object Does not See” means the
subject does not perceive the process as an Object)
SO #Segments under Object Does not See + #segments
under Object tries to See � #segments under
Object Sees
SO ! PO 0 < #segments under Object Does not See +
#segments under Object tries to See < #segments
under Object Sees
SO ! PO ! EO #Segments under Object Does not See ¼ 0 and
#segments under Object Tries to See ¼ 0 and
Object Sees includes exactly 1 context
Review descriptors and/or
transcript to determine
EO or O
#Segments under Object Does not See ¼ 0 and
#segments under Object Tries to See ¼ 0 and
Object Sees includes 2 different contexts
SO ! PO ! EO ! O #Segments under Object Does not See ¼ 0 and
#segments under Object Tries to See ¼ 0 and
Object Sees includes 3 or more different contexts
148 8 Totality as a Possible New Stage and Levels in APOS Theory
Object conception and not as a separate stage. In the last example, there is the
following comment:
We refer to an action when the subject is able to perform it in a step-by-step manner
but does not see it as a totality and cannot think about it, or explain it to another person.
(Cornu and Dubinsky 1989, p. 74)
While Cornu and Dubinsky state that the notion of totality is not part of an
Action conception, they do not propose totality as a new stage, separate from
Object, although they are somewhat vague about where totality fits relative to the
subsequent stages of Process and Object. In any case, they did not discuss the issue
beyond this single comment, nor did they make any investigation of it.
A series of papers (Weller et al. 2004; Dubinsky et al. 2005a, b) concerned with the
history of the concept of infinity inmathematics and the distinction between actual and
potential infinity, made several references to totality in statements such as:
If one becomes aware of the process as a totality, realizes that transformations can act on
that totality, and can actually construct such transformations, (explicitly or in one’s
imagination) then we say the individual has encapsulated the process into a cognitive
object (Dubinsky et al. 2005b, p. 256).
Brown et al. (2010) discussed totality as an important preliminary to encapsula-
tion, although they did not suggest it as a separate stage. There is also a mention of
totality in Dubinsky et al. (2008), again without considering it as a new stage.
Sfard (1992) also introduced a new stage between Process and Object. She called
it a condensed operational conception. By this she means the subject is able to see a
process as an input/output machine without paying attention to the details by which
an input is transformed into an output. This notion seems to be quite different from
the Totality stage discussed here. In any case, perspectives related to APOS Theory,
such as Sfard’s, may be discussed in a follow-up book.
The discussion of Totality in Dubinsky et al. (2013) introduced two main
differences with some previous studies that mentioned totality. First, as was
indicated above, previous studies generally considered totality as part of an Object
conception. Second, in Dubinsky et al. (2013), Totality is introduced as a possible
new stage supported by data-driven descriptions of the progression from Process to
Totality and from Totality to Object.
8.6 The Tentative Nature of Totality as a Stage
As of this writing, there is not sufficient evidence to decide whether Totality is
really any of the following: a stage, a level between Process and Object, a part of a
Process conception or a part of an Object conception. It seems clear that the Start,
Progress, and Emerging levels are only categorizations of the specifics in a
subject’s thinking at the time of the observation and so would be very different
for different concepts. Therefore they are best designated as levels. Based on the
research in Dubinsky et al. (2013), however, there is a strong likelihood that, at least
8.6 The Tentative Nature of Totality as a Stage 149
in the case of infinite repeating decimals, Totality functions as a separate stage as
opposed to being a level or part of another stage. This is because a Totality
conception appears to represent a change in how the individual thinks about the
mathematical concept. It may be that this development occurs in a similar manner
in the development of thought about other mathematical concepts. This seems
very likely for infinite processes and even possibly for finite processes that involve
a very large number of steps. In any case, the extent to which Totality appears as
a stage throughout the realm of mathematical concepts and the mechanism or
mechanisms by which a Totality conception is constructed (about which little or
nothing is known at present) remain matters for continuing research.
150 8 Totality as a Possible New Stage and Levels in APOS Theory
Chapter 9
Use of APOS Theory to Teach Mathematics
at Elementary School
Throughout the first half of the 1990s, the mathematics team of the Center for
Educational Technology, Tel-Aviv, Israel (CET), set out to revise the team’s
existing materials for teaching mathematics in Israeli elementary schools (Grades
1–6, ages 6–12). One important aspect of the revision was to introduce the ideas of
Piaget and APOS Theory into the teaching sequences. An area of particular interest
was the teaching of fractions in grades 4 and 5.
The data and conclusions described in this chapter emerge from two studies on
fractions: one on students’ understandings of part–whole relationships (Arnon
1998) and the other on students’ work with equivalence classes of fractions
(Arnon et al. 1999, 2001). They are described in this chapter in some detail in
order to acquaint the reader with the application of these ideas at the elementary
school level and to compare the learning of elementary school students with that of
postsecondary students.
Chapter 2 of this book describes how Dubinsky adapted Piaget’s ideas on
learning to postsecondary mathematics. According to Piaget, postsecondary
students are expected to be at the stage of formal operations. This means that the
objects on which they perform actions are abstract rather than concrete. In elemen-
tary schools, most students are at the stage of concrete operations. This means that
the objects on which actions are applied need to be concrete, that is, they can be
perceived by one’s senses (Piaget, 1975, 1974/1976). Thus, from the perspective of
APOS Theory, the principal difference between the elementary and postsecondary
mathematics classroom lies in the nature of the objects to which actions are applied.
The data presented in both studies and are discussed in this chapter were
collected in the early 1990s. APOS Theory has developed since then, including
the latest ideas about possible levels between stages and a possible new stage
between Process and Object (Totality, see Chap. 8). This chapter presents an
interpretation of the data collected in those studies in light of current APOS Theory,
which is somewhat different from the way it was originally presented in the 1990s.
In particular, this chapter includes use of the term level according to the meaning
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_9,© Springer Science+Business Media New York 2014
151
ascribed to it in Chap. 8: incremental points of progression in the development of a
concept between one APOS stage and its subsequent stage.
The chapter is organized as follows: The application of APOS Theory in the
elementary school versus its application at the postsecondary level will be
discussed in Sect. 9.1. Research about the learning of the part–whole interpretation
of fractions in grade 4 is discussed in Sects. 9.2–9.5. Section 9.2 consists of a
comparison of the performance of elementary school children who completed
APOS-based instruction on fractions with those who completed a unit on fractions
using a standard pedagogy that follows Nesher’s paradigm (Nesher 1989), which is
also Piagetian. Section 9.3 consists of a discussion of possible levels between the
Action and Process stages that Arnon (1998) identified in her investigation of
students’ thinking about some fraction concepts and the part–whole interpretation.
The role of imagination in the interiorization of an Action on concrete objects into
a Process conception is the subject of Sect. 9.4. Section 9.5 focuses on the concept
of equivalence classes of fractions in a fifth grade classroom. A summary of what
is known today about the use of APOS Theory in elementary schools is presented
in Sect. 9.6.
9.1 Applying APOS Theory in Elementary School Versus
Applying It in Postsecondary School
Piaget’s theory of cognitive development is based on the assumption that an
individual constructs knowledge by reflecting on her or his own experiences. At
the stage of concrete operations, these experiences need to be concrete (Piaget,
1975, 1974/1976). At the stage of formal operations, they can be abstract. The terms
concrete and abstract often have different meanings. Throughout this chapter they
will be used according to the meanings described in Arnon et al. (2001). Here, the
term concrete involves the use of real or imagined physical objects. For example,
many children, in dealing with fractions, feel comfortable when assisted by circle
cutouts (circle sectors) made of real cardboard paper, drawings, or mental
images (see Fig. 9.1). Other concrete representations of fractions are also described
Fig. 9.1 Circle cutouts
152 9 Use of APOS Theory to Teach Mathematics at Elementary School
in the literature. One can understand this interpretation of concreteness to mean that
a concrete experience needs to involve the learner’s senses.
In Fig. 9.1 one can see circle cutouts and whole circles. Some of the cutouts carry
the fraction they represent, some do not. Mathematicians also often use concrete
aids. For example, they often use Cartesian graphs when dealing with real-valued
functions and use drawings of two- and three-dimensional bodies when trying to
prove theorems in geometry (Arnon et al. 2001, p. 171).
The term abstract refers to the use of a mathematical concept without any
physical world representation (Arnon et al. 2001, p. 171). For example, people
often manipulate real-valued functions without any reference to their graphs or any
physical content that could be used to represent them; instead, they use mathemati-
cal language and syntax, as well as their knowledge of the mathematical structures
involved and the rules for combining their components.
The differences in APOS Theory between an abstract context (postsecondary
education) and a concrete context (elementary education) are illustrated in
Figs. 9.2 and 9.3.
Figure 9.2 shows the postsecondary context where the initial Objects, as well as
the Objects that emerge from the encapsulation of Processes, are typically abstract.
According to Piaget, children at the stage of concrete operations (approximately
from the age of 2 years up to 11 or 13 years) develop mental (abstract) concepts as
a result of reflection upon actions they perform in the real (concrete) world
with their own hands or in their imaginations (Piaget 1975, 1976). This is illustrated
in Fig. 9.3.
Figure 9.3 is a modification of Fig. 9.2 to represent the implementation of APOS
Theory for children at the stage of concrete operations. The figure illustrates how
Actions applied to physical Objects give rise to abstract mathematical Objects in a
child’s mind. At the stage of concrete operations, the Objects upon which the
Fig. 9.2 APOS for postsecondary students
9.1 Applying APOS Theory in Elementary School . . . 153
student acts need to be concrete. The Objects that emerge from the encapsulation of
the interiorized Actions are abstract, just as for postsecondary learners. Among the
participants of the study of Arnon (1998), two students demonstrated such encap-
sulation, as described in Sect. 9.3.3.
9.2 Comparing a Standard Instructional Sequence
to an Instructional Sequence Based on APOS Ideas
The grade 4 curriculum on fractions includes instruction on the part–whole inter-
pretation of fractions. For a fractionk
nk; n 2 Z; n 6¼ 0; an object that represents a
whole is divided into n equal parts, with k of those n parts selected.1 A circle and its
sectors (referred to as circle cutouts) were used as representations.
In a traditional teaching sequence, fourth grade students learned about the
part–whole interpretation by operating on circles as “wholes” and ready-made
circle cutouts representing a variety of unit fractions up to1
20, with n cutouts for
each fraction1
n. The appropriate fraction symbol
1
nwas printed on each of these
ready-made cardboard circle cutouts (Arnon 1998, p. 87). The following is an
example of an action operated on these manipulatives: students were asked to use
the separated cutouts to find out how many cuts of1
9were needed to fill an empty
circle or a cutout representing the fraction1
3. In this chapter, an instructional
Fig. 9.3 APOS for elementary school students
1 About the part–whole interpretation of fractions, and other interpretations, see in Arnon (1998),
pp. 65–74.
154 9 Use of APOS Theory to Teach Mathematics at Elementary School
sequence that used these types of materials is referred to as STN (for standard).Never in this instructional sequence did students actually divide a circle into equal
parts. As a result, it was suspected that these activities tended to lead the students to
reflect on the characteristics of the cutouts rather than the part–whole interpretation
they were meant to represent. Unofficial reports from schools suggested that some
of the fourth grade students who used these materials and activities developed good
part–whole conceptions; however, many did not.
The mathematics team set out to investigate how to improve this situation. The
ideas of Piaget and APOS Theory pointed to a direction.
Piaget distinguishes among three types of experience acquired through contact
with the external world:
A. “Simple exercise”, which “does not necessarily imply that knowledge will be
extracted. . .”B. “Physical experience”, where the child manipulates physical objects, and by
means of “a simple process of abstraction” abstracts the properties of the
objects.
C. “Logico-mathematical experience”, in which the child, manipulating objects,
constructs properties of the action itself and of the transformation(s) he or she
applies to the objects. In this type of experience knowledge is constructed by
means of reflective abstraction, as described in Chap. 2. (see also, Piaget 1975,
p. 193–194)
In Sect. 2.2 we have seen examples of how Piaget’s theory of reflective abstrac-
tion formed the antecedents to APOS Theory—the mental structures as well as
mental mechanisms such as interiorization (Dubinsky 1991). Yet the activities of
the STN sequence are more like Piaget’s Type B experience. As a result, instead of
a reflective abstraction and interiorization of part–whole Actions, the students used
“simple processes of abstraction,” or Type B experience, and tended to identify
properties of the circle cutouts.
A new teaching sequence was developed, the goal of which was to increase the
likelihood of Type C experiences and to enhance the construction of a part-whole
interpretation of fractions. To that end the development team introduced a different
set of materials, which started with a cardboard page illustrated in Fig. 9.4. This
cardboard tool was called partitioning rings.
Fig. 9.4 The partitioning
rings (Arnon 1998, p. 210)
9.2 Comparing a Standard Instructional Sequence to an Instructional Sequence. . . 155
In addition to the partitioning rings, the manipulative set included drawings of
preprepared empty circles of the same size, with center points. The inner circles of
the partitioning rings were perforated to be easily pushed out. To construct a
representation of a fraction, sayk
n, the student had to choose a ring that was divided
into n equal parts, place it on an empty circle that showed the center point, and use
the segments indicated on the outer ring to divide the circle into n equal parts and
then shade k of those parts. This was the action the students were expected to
interiorize. This action consisted of the following steps: For a fractionk
n, one
chooses a circle (“the whole”), divides it into n equal parts (according to the
denominator), and finally shades k of these parts (according to the numerator).
Later in this instructional sequence, ready-made circle cutouts were introduced in
class, representing a variety of fractions, including non-unit fractions
k
nwith k > 1
� �. These cutouts were purposely left unnamed since an important
activity was to have the students name them (Arnon 1998 p. 87–89). This instruc-
tional sequence, as well as the class where it was used, is referred to in this chapter
as EXP (for experimental).Since the use of the ready-made cutouts encourages Type B rather than Type C
experience, APOS Theory would suggest that the introduction of ready-made
cutouts should be delayed until the action of drawing circle representations of
fractions is interiorized. The main evidence that such interiorization has taken
place is the student’s ability to describe verbally how to produce such a representa-
tion. The following excerpt from the interview with Offir provides an example.
With none of the concrete materials present, Offir was asked to compare1
100and
1
13, neither of which was included in the set of concrete materials. In his response,
he explains why he thinks that1
100is smaller than
1
13:
. . .Now, when you divide the whole into thirteen, the parts which are narrow, are of some
specific size, and when. . .eh. . ., when you divide the whole into one over a hundred, so thedensity is that,. . .there is more density, and the part of the hundred, the circle-cutout of the
hundred becomes more, eh, smaller [showing with a gesture of two fists the act of
narrowing, of getting closer]. (Arnon 1998, p. 105.)
Another indication of interiorization is the student’s ability to produce a sche-
matic drawing of a fraction without using the partitioning rings. This is illustrated in
Figs. 9.5 and 9.6, where one can see two hand drawings performed without the
tools; Yulia makes the first drawing and Gali the second:
156 9 Use of APOS Theory to Teach Mathematics at Elementary School
Despite the clumsiness of the productions, the correct ideas are present.
Drawings of this kind demonstrate a student’s ability to reconstruct physical
representations when asked to perform specific actions on fractions.
Both learning sequences, EXP and STN, were based on Piaget’s idea that the
construction of a new mathematical concept begins with actions applied to physical
objects (Piaget, 1975; 1974/1976). The difference between the approaches lies in
the nature of the actions used in each instructional sequence. Arnon (1998) set out
to investigate the question of whether the EXP instructional sequence led to
improved learning. The data showed that it did.
The following four fraction concepts were investigated in Arnon’s (1998) data
analysis: comparison of unit fractions, construction of non-unit fractions, compari-
son of non-unit fractions, and multiplication of a unit fraction by an integer. As for
the arithmetic operations, the two sequences had an important common trait: their
way of treating the arithmetic operations of addition, subtraction, comparison, and
multiplication. These operations were not meant for the learning of the algorithms,
but only to serve as catalysts to encourage encapsulation, so that the students’
conceptions of fractions would develop from Process to Object. In both sequences
students only practiced cases of these operations that could easily be solved with the
aid of the manipulatives. For example, addition problems were presented only if
they involved cutouts that existed in the set of manipulatives and where one
Fig. 9.5 Approximate
drawings by Yulia (Arnon
1998, p. 220)
Fig. 9.6 Approximate
drawings by Gali (Arnon
1998, p. 115)
9.2 Comparing a Standard Instructional Sequence to an Instructional Sequence. . . 157
denominator was a multiple of the other. For a problem such as2
5þ 1
10, the students
would arrange the proper cutouts next to each other, then cover the2
5with
1
10
cutouts, and conclude with the solution3
10. No computations were involved.2
Two classes, one STN and one EXP, participated in the study. Instruction for
each class took place during the last trimester of grade 4. The students were
interviewed when they reached grade 5, after summer vacation. The interviews
were individual, audio recorded, and transcribed. The physical objects that were
used in Grade 4 were not present in the interviews. They were only referred to
verbally, and students often reconstructed them in the drawings they made during
their interviews. The interview protocol consisted of the following types of arith-
metic problems: comparison of two unit fractions, comparison of two non-unit
fractions, and multiplication of a unit fraction by an integer. Information about the
concepts constructing a unit fraction and constructing a non-unit fraction was
derived from the discussions of the binary operations.
The data provided information about the following issues: comparative mathe-
matical performance between the two groups, STN and EXP (in this section); the
existence of possible developmental levels between Action and Process for the four
investigated concepts (Sect. 9.3); preliminary genetic decompositions for the same
four concepts with a detailed description for the concept construction of a non-unitfraction (Sect. 9.3); and the prevalence of operating on concrete objects in the
children’s imagination (Sect. 9.4).
In analyzing students’ progression from Action to Process, responses appeared
to fall into one of three categories:
i. Not further than an Action conception: No evidence of interiorization of any
Action on concrete objects
ii. The transition from Action to Process: Evidence of interiorization of either a
partially or completely incorrect Action on concrete objects
iii. At least Process conception: Evidence of interiorization of correct Actions
Because the manipulatives were not used in the interviews, the researcher could
only detect levels of interiorization, and not knowledge of the Action itself. But she
could trace No Evidence of Action by the lack of a description or drawing that
referred to either fractions or concrete representations of such or by lack of any
response. Such evidence was categorized as No Evidence of Action (category i).3
The author could also trace verbalizations and/or drawings that indicated the
2According to the curriculum, in grades 5 and 6, where their fraction conceptions hopefully have
developed into Objects, they will learn the proper algorithms for arithmetic operations of fractions.3 In cases where the student provided a No evidence of action response, it might be the case that
provided the manipulatives, the student might perform a correct Action.
158 9 Use of APOS Theory to Teach Mathematics at Elementary School
interiorization of a partially or completely incorrect action, which indicated that
the student’s conception was in transition from Action to Process. Such responses
were categorized as category ii. Responses that revealed the interiorization of a
complete and correct Action were categorized as category iii, indicating a Process
conception.
Figure 9.7 illustrates the interiorization of a completely incorrect action (cate-
gory ii) for the construction of a non-unit fraction by Avi:
In the interview Avi produced this drawing to represent3
5(Arnon 1998, p. 111)
and said the following:
I: How many fifths do we see in your drawing?
. . .
Avi: Ah, so in every circle there are five, and if we join them all we have fifteen.
I: So three fifths is fifteen fifths?
Avi: This [the drawing]? five over, eh, three fif. . .,three over.. . .
I: Explain again, why did you do here three circles?
Avi: Because each one has, eh, because it is written in the numerator three.
Avi’s action was incorrect in two ways: first, he misinterpreted the numerator by
thinking that it could be represented by three circles, and then, he divided each of
the three circles into five equal parts and shaded all 15 parts. None of these action
steps were correct. Avi’s response was judged as category ii.
The following example illustrates the interiorization of an action that was only
partially incorrect (category ii). Dafni constructed a unit fraction. Then she
explained what a student should do in order to construct the unit fraction1
10:
Fig. 9.7 Avi’s incorrect action for construction of the fraction3
5
9.2 Comparing a Standard Instructional Sequence to an Instructional Sequence. . . 159
Dafni: In all the circle there ought to be ten. And after that he erases one [Author’s emphasis].
Dafni’s suggestion to divide the circle into ten equal parts, shade one, and leave
nine unshaded was correct. Her call to erase the shaded portion was not, because it
indicated her belief that the unshaded parts represent the fraction. Dafni further
exhibited her incorrect conception of non-unit fractions in her attempt to compare2
5
with4
6:
I: Do you want to explain to another child what he should do in order to check [the
comparison]?
Dafni: He needs to draw the, eh, nu, the, eh, circle of six, six parts, and color there four.
I: Yes.
Dafni: After that he knows what is left of it, and after that he does the same thing with two
over five.
I: Yes? And, how does one know which is greater?
Dafni: According to,.. According to the size that came out.
I: According to the size of what he colored or according to the size of that which was left?
Dafni: Of that which was left.
Similar to her representation of1
10, Dafni again divided the circle correctly and
shaded the right number of portions. However, she discarded the shaded portion and
misinterpreted the fraction as “that which was left” unshaded. This again
constituted a partially incorrect action (category ii). This was the case with Dafni
for each of the concepts investigated: comparison of unit fractions, constructing anon-unit fraction, comparing non-unit fractions, and multiplication.
For each of the four concepts investigated in the study, the three categories of
interiorization accounted for all of the students’ responses. When comparing the
achievement of the STN and EXP students for each category and for each concept,
the EXP class always did better. Table 9.1 illustrates the comparative results (STN
versus EXP) for the comparison of non-unit fractions. Notice that for this concept, acorrect action consisted of the following steps: take the two appropriate cutouts andput them one upon the other so that they are either equal or one is completelycovered by the other. If the cutouts are equal then the fractions are equal; if onecutout covers completely and overflows the other, then the fraction it represents isthe larger fraction. This action is difficult to accomplish without the cutouts;
nevertheless about a third of the STN students and two thirds of the EXP students
succeeded in solving it.
Table 9.1 Comparing the percentage of students of each class for the concept comparing non-unit
fractions
Comparing non-unit fractions, degree of interiorization
Class STN
N ¼ 28 (%)
Class EXP
N ¼ 32 (%)
Not further than Action conception 32 12.5
Transition from Action to Process 39 22
At least Process conception 29 66
160 9 Use of APOS Theory to Teach Mathematics at Elementary School
A comparison of the results in Table 9.1 shows that the EXP class fully
interiorized the action of comparing non-unit fractions at more than twice the rate
of students from the STN class. Comparative results were similar for all four
concepts that were investigated in the study. A cumulative comparison is shown
in Table 9.2 (Arnon 1998, p. 174):
Table 9.2 shows the percentages of students in each class who correctly
interiorized exactly none, one, two, three, or four of the actions. For example, for
the four concepts studied, 43 % of the EXP students versus only 21 % of the STN
students were judged to have interiorized all four correct actions, namely, achieving
at least Process conceptions for all of the investigated concepts; 70 % of the EXP
versus 39 % of the STN students interiorized actions for more than two of the
investigated concepts. As the table shows, the EXP students made correct interior-
izations more readily than the STN students. This reinforces the claim that the
action of constructing fraction representations with the help of the partitioning rings
(the one used in the EXP class) was more easily interiorized than activities with the
ready-made (and named) circle cutouts (as used in the STN class). Two of the EXP
students even gave evidence of achieving Object conceptions of the concept
constructing non-unit fractions, while none of the STN students provided such
evidence (see Sect. 9.3.3).
9.3 Levels and Genetic Decompositions for the Transition
from Action to Process of Some Fraction Concepts
In this section the term level is used in the sense elaborated in Chap. 8. Level refers
to developmental phases that occur in the transition from one APOS stage to
another.
9.3.1 Levels in the Developments of Some Fraction Concepts
In the study of elementary school students’ learning of fractions, Arnon (1998)
concentrated on the transition from Action to Process. For Piaget this transition was
from an unconscious application of the action to full consciousness of it. In his
Table 9.2 Overall interiorization
Overall interiorization
STN class
N ¼ 28 (%)
EXP class
N ¼ 32 (%)
No proper actions interiorized 25 17
Exactly one proper action interiorized 21 10
Exactly two proper actions interiorized 14 3
Exactly three proper actions interiorized 18 27
Exactly four proper actions interiorized 21 43
9.3 Levels and Genetic Decompositions for the Transition from Action. . . 161
book The Grasp of Consciousness, Piaget (1974/1976) discusses the gradual
development of consciousness. According to Piaget, that passage “must require
constructions, and cannot be reduced to a simple process of illumination” (Piaget
(1974/1976), p. 322). Formany of the situations he investigated, Piaget determined the
existence of incremental points of progression from the unconscious to the conscious.
Phenomena similar to what Piaget described are discussed in Chap. 8. Such
incremental points are named in APOS Theory as levels. In the development of the
concept of repeating decimals, the researchers identified levels of development
between every two consecutive stages of APOS, including a proposed new stage,
Totality (Dubinsky et al. 2013). In her study of different fraction concepts, Arnon
(1998) identified levels in the transition from Action to Process.
Similar to what Piaget found and what is reported in Chap. 8, Arnon (1998)
determined that levels, unlike stages, are not general but are content specific. Levels
for the concept constructing a non-unit fraction are described in Table 9.3.
For the transition from Action to Process of this concept, the researcher showed
the existence of two levels, which are given in the right-hand column of Table 9.3.
Levels for the other investigated fraction concepts were also found. For example,
for the concept of comparing two non-unit fractions, six different levels were
identified on the transition from Action to Process. These levels appear in Table 9.4
(Arnon 1998):
Notice that identifying a student’s conception as being at some level within the
transition from Action to Process does not imply that the student necessarily went
through all the previous levels. He or she might have skipped some of them. More
research is needed to establish the existence of levels in this case.
Table 9.3 Levels in the interiorization of constructing a non-unit fraction
Constructing a non-unit fraction
The transition from
Action to Process
� Drawing k (numerator) circles, each divided into n (denominator)
equal parts—a completely incorrect action
� The rest (shading ¼ erasing)—only part of the action incorrect
Table 9.4 Levels in the interiorization of comparing two non-unit fractions
Comparing two non-unit fractions
The transition from Action to Process � Larger numerator ) larger fraction (counting
circles)
� Larger denominator) larger fraction (counting
equal parts)
� Smaller integers (numerator and denominator), mean
larger parts, and hence a larger fraction
� Smaller denominator means larger parts, and hence
a larger fraction
� Larger non-shaded area ) larger fraction
� Larger number of non-shaded parts ) smaller
fraction
162 9 Use of APOS Theory to Teach Mathematics at Elementary School
9.3.2 Genetic Decompositions for the ConceptsThat Were Investigated
The data in the tables for the different concepts could be interpreted as levels in the
transition of each concept from Action to Process. A preliminary genetic decompo-
sition arising from each of the tables for each of the concepts that were investigated
can be useful in continuing research, as described in Asiala et al. (1996) and in
Chaps. 4 and 6.
Figure 9.8 presents one such genetic decomposition (for the transition from
Action to Process) derived from Table 9.3 for constructing a non-unit fraction.The findings described above contain also some data about the transition from
Process to Object, yet not enough to design a full genetic decomposition. Neverthe-
less, some interesting findings about an Object conception will be presented in
Sect. 9.3.3.
9.3.3 Additional Achievements: Abstract Objects
As indicated above, both the EXP and STN instructional sequences included simple
arithmetic operations and comparisons of fractions that were performed with
physical manipulatives. These actions were intended to support the encapsulation
of the conception of fraction as Process into the conception of fraction as mental
Object. The students were not taught any algorithms. Although the study set out to
investigate the development from Action to Process, two students gave evidence of
having reached the Object stage when they performed formal actions of comparison
on fractions as abstract objects. For example:
Judi, an EXP student, used1
2as an abstract benchmark when comparing
2
5and
4
6:
I: What about two fifths and four sixths, which is larger?
Judy: It seems to me that four sixths.
I: Why?
Judy: Because two fifths is smaller than a half, and four sixths is already more, because three
sixths is a half, and two fifths has not yet reached a half. (Arnon 1998, p. 129)
Byway of comparison, Yulia also used1
2as a benchmark for comparing
4
10and
3
4:
Yulia: Because four tenths is approximately such a thing [gesture of an arc smaller than 180�]and this will be approximately like this. . . more [gesture of an arc larger than 180�].(Arnon 1998, p. 155)
Fig. 9.8 A genetic decomposition from Action to Process for constructing a non-unit fraction
9.3 Levels and Genetic Decompositions for the Transition from Action. . . 163
Unlike Judi, Yulia referred to1
2with gestures of imaginary arcs, smaller and
larger than 180�. Comparing these two excerpts highlights Judi’s response as
evidence of having constructed an Object conception of fractions.
Dan, also an EXP student, used fraction equivalence and transitivity to explain
why4
6was larger than
2
5:
Dan: 2
5is
4
10, and
4
6is larger than
4
10. (Arnon 1998, p. 129)
Dan and Judi, both EXP students, appeared to operate on the non-unit fractions
as abstract objects. It seems that they had encapsulated the Process into an Object.
They solved the comparison problem properly and made no reference to concrete
objects. Although they started their learning with actions on concrete objects, they
constructed abstract objects, as predicted by APOS Theory. No student in the STN
class gave evidence of this type of development.
9.4 Manipulating Concrete Objects in the Imagination
For Piaget, the interiorization of actions on concrete objects is demonstrated by the
emergence of the learner’s consciousness of the actions. This is described in detail
in The Grasp of Consciousness (Piaget, 1974/1976). In APOS Theory, interioriza-
tion is described as the emergence of an individual’s ability to gain internal control
over the Action. For learners at the stage of concrete operations, this, according to
APOS Theory, might be evidenced by the learner’s ability to carry out the Action in
her or his imagination. Also, the levels that emerged from the analysis described
above were defined in terms of Actions operated in the imagination on imaginary
concrete Objects. In order to understand the scope of the newly defined levels, it
was interesting to investigate the prevalence of manipulating concrete Objects in
the imagination.
In the interviews (Arnon 1998), students completed tasks they originally
learned to perform with manipulatives. However, the interviews took place
without manipulatives. One of the purposes of the study was to collect data
about the methods the students used for solving such problems in the absence
of their manipulatives.
Some of the students produced drawings that approximated their use of
manipulatives. The ability to produce such drawings suggested that the student
had performed the Action in her or his imagination. Other subjects used
terminology and gestures that indicated evidence that they carried out these
Actions in their imaginations. Arnon developed six criteria to indicate instances
of a learner using her or his imagination to carry out an action on imaginary
concrete objects:
164 9 Use of APOS Theory to Teach Mathematics at Elementary School
9.4.1 Criterion 1: The Student Declared Explicitly Thatthe Answer He or She Had Provided Was a Resultof Actions Which He or She Had Performedon Imaginary Concrete Objects
Following are some examples.
EXP student Gil:
. . .if I draw two fifths, then I’ll have. . .eh,. . .eh,. . .a third?. . .Because four sixths, a sixth
and a sixth, is two sixths, and two sixths equals a third. So it is as if two thirds. (Arnon 1998,
p. 128)
Gil’s words, “if I draw two fifths,” constitute a declaration she imagined, or
could imagine, drawing a representation of2
5. Yet no drawing was present in the
interview.
EXP student Roni said that4
6was larger than
2
5. In his explanation he declared
that he had used an imaginary drawing:
I: How do you know?
Roni: Ah,. . .I did the circle,[no drawing present] and it came out more,..mm,..in the
comparison it came out bigger.
I: You did the circle in your head?
Roni: Yes. (Arnon 1998, p. 151)
In the next excerpt, EXP student Lina tries to explain why she thinks2
5is larger
than4
6.
Lina: A fifth is a bigger part. Two fifths and the,. . .a sixth.I: How are you trying to find out? I can see that you are thinking.
Lina: . . .I: What are you trying to do in your head in order to know?
Lina: To see what each form looks like, to arrange it.
In Lina’s last response, she gives evidence of constructing a representation
(“what each form looks like”) and of carrying out an Action in her imagination
(“to arrange it”).
Her use of imagination continued to be apparent in the following excerpt:
I: Maybe you would like to draw it?
Lina: No.I: No? You don’t have to? Then what do you draw in your head? What do you draw in your
head? Describe to us what do you try to, to arrange in your head?
Lina: The circle.
I: Yes. . .Lina: How the circle,..together. . .
I: What does the circle that. . ., what does your drawing look like?
Lina: You divide into five parts
9.4 Manipulating Concrete Objects in the Imagination 165
I: Yes, and
Lina: And,.. one takes two fifths, yes.
I: What do you mean? What does one do? You took a circle and divided into five parts.
What do we do now?
Lina: Eh,.. the second circle, divide into six.
I: Still with the first circle. What do you do with the first circle in order to see two fifths.
Lina: . . .I: Is it enough to divide it into five parts?
Lina: No.
I: But?
Lina: . . .I: What do you do?
Lina: To color?
I: To color? What do you color?
Lina: Eh, two, two fif, eh, two fifths.
I: O.K. Now, what do you do in the second circle?
Lina: You divide into six parts, six parts, and color four sixths.
Since no drawing was present throughout the entire conversation, the researcher
judged this to fall under Criterion 1 (Arnon 1998, p. 151).
9.4.2 Criterion 2: Activating Imaginary Circle CutoutsThat Did Not Exist in the Original Set of Manipulatives
Offir’s comparison of1
100and
1
13, which was discussed in Sect. 9.2, is an example
of a Criterion 2 response. This idea coincides with Piaget and Inhelder’s (1966/
1971) notion of authentic anticipatory image, which occurs when the subject has toimagine an object unknown to her or him in advance and anticipate actions applied
to that object. In the following excerpt, Sharon, an STN student, appears to offer
authentic anticipatory imagery:
I: . . .How, if you had these hard-paper objects, how would you use them to check [the
answer that1
7was larger than
1
11?]
Sharon: I’d see, let’s say, one, eleven, one over eleven, eh, we see that it is so small, even in the
classroom, even with that thing.
I: Did you have one over eleven in the classroom?
Sharon: Eh, no, there was not one over eleven
I: Yes, so how do you know about one over eleven?
Sharon: . . .I: Was there one over eleven in the classroom?
Sharon: No.
I: So how do you know about one over eleven if it was not in the classroom?
Sharon gives evidence of authentic anticipatory image in her response to the
interviewer’s question:
Sharon: Ah, there was also a one over ten. One over ten is close to eleven. So we checked with
the ten, and we also saw with six, with that thing, which was bigger.
166 9 Use of APOS Theory to Teach Mathematics at Elementary School
She provides additional evidence somewhat later:
I: How did you check?
Sharon: We put the one over eleven, and we put the one over seven like that, above it, like that
and you check. If you have a space left of the one over seven, you know that the one
over seven is bigger. (Arnon 1998, pp. 153–154)
9.4.3 Criterion 3: The Use of Drawings
The students’ drawings were necessarily inaccurate approximations of the real
manipulatives. In the case of comparing fractions, inaccurate drawings by them-
selves were of little use, because of the way comparison was carried out in class: put
one circle cut upon the other to see which is physically larger (see Sharon’s descrip-
tion of such comparison in the last part of her excerpt). Arnon describes conditions
for drawings to be counted as evidence of performing an action in one’s imagination.
Rikki, an EXP student, made useful drawings that constituted evidence of Criterion
3. In the following example, Rikki tries to solve amultiplication problemby using the
drawing she provided. In her effort to solve the problem1
5� 3, she makes a drawing
to represent1
5(Fig. 9.9):
Then she proceeds verbally, describing a mental operation:
Now it is, eh,..as if multiplied by three, so it is three, eh,..it is two more like this, as if, as if it
equals three fifths? (Arnon 1998, pp. 155–156)
9.4.4 Criterion 4: Verbal Indications That Involve the Useof Language That Refers to the Concrete Manipulatives
Arnon (1998) provides a long list of expressions that were accepted as indicators of
an individual manipulating concrete objects in her or his imagination (p. 160).
A special case of verbal indicators was use of the terms “part” and “the whole.”
These expressions do not necessarily indicate concrete connotations. Use of these
Fig. 9.9 Rikki’s drawing of1
5
9.4 Manipulating Concrete Objects in the Imagination 167
terms was considered an indication of an Action carried out in one’s imagination
only if somewhere in the interview, there was evidence of the student using the terms
“part” and “the whole” to symbolize concrete entities. An example follows.
Effi, an STN student, when explaining his comparison of unit fractions, used the
terms parts and the whole with no indication of concrete meaning. Yet, later in the
interview, when comparing two non-unit fractions, he said:
Effi: the two and the four are, we learned that the four one should color out of the six parts, and
out of the five one should color two parts. (Arnon 1998, pp. 156–157)
The use of the expression “color two parts,” although it came in a different part
of the interview, ensured that for this student, “parts” were concrete objects.
9.4.5 Criterion 5: Gestural Indications
Following are some examples of gestural indicators. In the following excerpt, Yulia
uses hand gestures to represent fractions larger and smaller than a half:
I: Which is larger4
10or
3
4
� �?
Yulia: Three quarters.
I: Why?
Yulia: . . .I: How do you know?
Yulia: Because four tenths is approximately such a thing [gesture of an arc smaller than 180�]and this will be approximately like this,. . . more [gesture of an arc larger than 180�].(Arnon 1998, p. 155)
Maya, an EXP student, uses gestures in her construction of one seventh1
7
� �:
Maya: One does a whole [with her finger ‘draws’ a circle in the air].
. . .
Maya: One divides it into seven parts[with her finger ‘draws’ radii of the imaginary circle in the
air]. (Arnon 1998, p. 158)
Criteria 1–5 correspond to three of the criteria distinguished by Piaget and
Inhelder to identify Actions in the imagination: verbal expressions, drawings, and
body gestures (Piaget and Inhelder 1966/1971). In the present study, an additional
criterion was used, as described below.
9.4.6 Criterion 6: Prompting
Often in the interview the interviewer encouraged the student to refer to concrete
Objects used in class. Arnon called this type of interference prompting. Evidence ofoperating imaginary Actions upon imaginary Objects was counted according to the
168 9 Use of APOS Theory to Teach Mathematics at Elementary School
different chronological relations between the evidence and the prompt (Arnon
1998). For more about the role of prompts in APOS-based research, see Chap. 6.
Altogether, 44 students, or 70 % of the interviewees (of both EXP and STN
classes), provided evidence of manipulating concrete Objects in their imaginations
when solving formally presented problems, an indication that they had interiorized
these Actions. Also, all the students who interiorized correct Actions for all four
concepts of the experiment provided such evidence (Arnon 1998).
These results support the role of imagination in the transition from Action on
concrete Object to the development of a Process conception. When this Process is
itself encapsulated, the resulting Object is an abstract Object.
Also, the high percentage of students who operated on concrete Objects in their
imagination gives hope that the levels defined above have some generality.
One can find evidence in the literature of similar behavior by adults. For
example, Hatano et al. (1977) studied the performance of expert abacus users and
found that users of intermediate skill employed imitative finger movements when
solving problems without an abacus, while advanced users testified to having used
finger movements earlier, but no longer needed to do so. Similar behavior was
exhibited by Arnon’s “experts”: all (100 %) of the students who interiorized correct
Actions for all four concepts of the experiment provided evidence of performing
concrete Actions in their imaginations.
9.5 Equivalence Classes of Fractions in Grade 5
This section discusses an experiment designed to investigate the use of APOS
Theory in the teaching of formal concepts of advanced mathematics at the elemen-
tary school level, in this case the concept Fractions as Equivalence Classes4 in
grade 5 (Arnon et al. 1999, 2001). A software program was developed specifically
to serve as a concrete (graphical) representation for the teaching of this concept.
This representation consists of points and lines in a discrete Cartesian coordinate
system, which will be presented in detail in Sect. 9.5.2.
This study is an APOS-based study in the following sense: the learning of the
mathematical concept began with an Action on concrete Objects (drawings); the
learning sequence consisted of small group activities, class discussions, and addi-
tional exercises; 20 (out of 30) students participated in individual interviews after
the instruction; the interview data was analyzed using APOS Theory; the software
and the learning sequence were constructed so that students first learned to con-
struct representations of fractions, classes, and binary operations in a step-by-step
manner, and only after interiorizing these constructing Actions did they start
working with ready-made representations (similar to the learning sequence of the
4 See the appendix at the end of this chapter for the definition of fractions as equivalence classes
and Q(R), the quotient field of a commutative ring.
9.5 Equivalence Classes of Fractions in Grade 5 169
EXP class in the study of fractions); and the algorithms for the binary operations
were not taught in class. The operations were used to enhance encapsulation of
Process conceptions to Object conceptions. In the search for evidence of Process
conceptions of Equivalence Classes, Arnon et al. (1999, 2001) found situations
where students used imaginary lines and points when solving arithmetic problems
such as comparisons, again similar to the findings of the fraction study described in
Sects. 9.2–9.4.
9.5.1 Equivalence Classes of Fractions in the Literature
Several studies have dealt with middle grade students, preservice teachers, and
university mathematics students’ difficulties in learning about equivalence classes
(e.g., Asghary and Tall 2005; Chin and Tall 2001; Hamdan 2006; Mills 2004).
Moreira and David (2008) claim that the study of fractions as equivalence classes is
important for prospective teachers’ understanding of the real numbers, even though
it is not part of the school syllabus. Although Chin and Tall (2001) report on the
representations of equivalence classes by means of points and lines on a discrete
grid, they did not use it in their teaching because they estimated that it was too
complex, even for postsecondary students. Arnon et al. (1999, 2001) report on a
teaching experiment with 5th graders, who learned about equivalence classes of
fractions using this representation.
9.5.2 The Experiment
Software was designed to provide students with a concrete environment in which
they could work with visual representations of fraction-related concepts.
The environment consists of a Cartesian coordinate system, where a single fractiona
bis represented by a discrete point on the system, with the denominator
b represented on the horizontal axis and the numerator a represented on the verticalaxis. The origin, as well as the entire vertical axis, is inaccessible (because the
denominator is 0).
An example screen appears in Fig. 9.10. Formal mathematical expressions are
on the left (here a green isolated fraction2
3and the two equivalence classes: for
1
5,
given in yellow, and1
3, given in purple), and each numeric object is represented,
with its corresponding color, by a graphical representation on the right.5 A fraction
appears as a discrete point on the grid, and its equivalence class is represented by a
line that passes through the origin and the point, passing through all and only the
points of the equivalence class. During the first part of the learning sequence, the
5 The colors are displayed in the electronic version of the book.
170 9 Use of APOS Theory to Teach Mathematics at Elementary School
numeric part of the screen is inaccessible to the student. The students work in the
graphics window (the right-hand side of the window). Here they can construct
points that represent fractions and lines that represent equivalence classes, and
watch the corresponding arithmetic expression appearing automatically on the left-
hand side of the screen following their constructions. The graphics window also
allows them to compare fractions and perform arithmetic operations. The software
and activities were designed to help the students see that arithmetic and compara-
tive operations are independent of the equivalence class representatives selected.
Similar to the instruction sequence of the study of fractions with circle cutouts(Arnon 1998), arithmetic operations were not taught using algorithms. Rather,
computer activities involving comparison, addition, and subtraction were meant
to encourage encapsulation of the Process of forming equivalence classes.
Although the software can also include fractions with negative denominators or
numerators, the 5th grade students involved in this study worked only with fractions
with positive denominators and nonnegative numerators.6
Thirty 5th graders (ages 11–12) participated in the experiment. When the
instruction sequence was over, 20 of these students participated in audio-recorded
individual interviews. The interview problems were presented to them in either
formal language or drawings. Only after solving a problem without the computer
was the interviewee allowed to check her or his result with the software. The study
describes in detail students’ responses and behavior in these interviews and
analyzes them according to APOS Theory.
The data reveal that most of the interviewees developed an Action or Process
conception or were in transition from Process toward Object conception of equiva-
lence class.
In the instructional sequence, students solved traditional fraction problems
(addition, subtraction, and comparison) using equivalence classes and the software
tools instead of numerical algorithms. This is illustrated in examples of student
work given in Figs. 9.11 and 9.12.
Fig. 9.10 The concrete environment (Arnon et al. 1999, p. 35)
6 The representation of fractions described above was also dealt with by Kalman (1985), Kaput and
Hollowell (1985), Kieren (1976), and Lemerise and Cote (1991).
9.5 Equivalence Classes of Fractions in Grade 5 171
Fig. 9.11 Dora, a
low-achieving student,
solved an addition problem
with different members of
the relevant equivalence
classes
Fig. 9.12 Limor, an above average achieving student, solved an addition problem with different
members of the relevant equivalence classes
This experiment established the plausibility of teaching the concept equivalenceclasses of fractions in elementary schools and perhaps other advanced mathemati-
cal concepts. As for this concept, more research is needed to establish a preliminary
genetic decomposition that will serve as a beginning of a series of research studies
that will produce an adequate genetic decomposition according to the APOS
framework, described in Asiala et al. (1996).
9.6 What Is Known About the Use of APOS Theory
in Elementary School
Following is a summary of what was learned from the studies reported in this
chapter about the use of APOS Theory in elementary school.
Children did better in developing a Process conception of fractions when starting
with an Action of producing a fraction representation that corresponds to the
part–whole interpretations of fractions than children who started with Actions on
ready-made concrete representatives. For example, in the fractions study discussed
in Sect. 9.2, only 27 % of the STN students interiorized proper Actions for all four
concepts studied versus 43 % of the EXP students. The study (Arnon 1998) showed
that the Action of producing concrete representations of a mathematical concept is
more effective than Actions on ready-made representations.
In the case of fractions, there might be several levels (substages)7 in the
transition from Action to Process. For example, the data showed the existence of
two such levels—interiorization of a completely incorrect action and interioriza-tion of a partially incorrect action—between the Action and Process for the concept
non-unit fraction. As is typical to levels, they are specific to each concept.
A criterion for the interiorization of actions performed on concrete Objects was
established: when shifting from Action to Process or within the Process stage, the
learner performs the Action in her or his imagination when solving problems
presented formally. The data showed that 70 % of all the interviewees of Arnon
(1998) (both EXP and STN students) provided evidence of that.
In Sect. 9.3 criteria were suggested for determining when an Action takes place
in the imagination: by body gestures, use of language, or use of approximate
drawings [all of which were found in both studies, that of elementary fraction
concepts and that of fractions as equivalence classes (Arnon 1998; Arnon et al.
1999, 2001)]. These criteria may be useful in future research.
Advanced mathematical concepts, such as equivalence classes of fractions, can
be adapted to the elementary school level by means of appropriate concrete Objects
and adequate Actions. APOS Theory was used to construct a teaching sequence that
enabled the students to develop meaningful conceptions of the topics in question.
7 About the difference between level and stage, see Chap. 8.
9.6 What Is Known About the Use of APOS Theory in Elementary School 173
Appendix: Fractions as Equivalence Classes: Definition
Let R be a commutative ring without zero divisors.
(a) We define a relation on R� R n 0f gð Þ by ða; bÞ � ðc; dÞ , ad ¼ bc.
This is an equivalence relation. The equivalence class of ða; bÞ is denoted by a
b.
(b) The set QðRÞ :¼ a
bj a 2 R; b 2 R n f0g
n oof equivalence classes,
endowed with the operationsa
bþ c
d:¼ ad þ bc
bdand
a
b� cd:¼ ac
bd; is a field,
called the quotient field of R (Spindler 1994, V. II, p. 40).
174 9 Use of APOS Theory to Teach Mathematics at Elementary School
Chapter 10
Frequently Asked Questions
This chapter consists of answers to questions about APOS Theory that either have
appeared in print or have arisen in personal communications with the authors. The
format for this chapter is similar to that of an interview: there is a question or
statement followed by a response from the authors. Where appropriate, the response
will include a reference to one or more of the chapters in this book.
10.1 Questions About Structures, Mechanisms,
and the Relationship between APOS Theory
and the Work of J. Piaget
Q: Is a Process a generalization of the Action to which it corresponds?
A: A process is not a generalization; it is a reconstruction of a transformation of
Objects onto a higher plane—from the plane of external, physical trans-
formations (Action) to the plane of mental transformations (Process). In this
sense reconstruction of a transformation refers to the learner’s ability to inter-
nalize an external Action through the mechanism of interiorization so that the
transformation is wholly under the learner’s control.
Q: What is the difference between a mental structure and a mental mechanism?
A: A mental structure is any relatively stable (although capable of development)
transformation that an individual uses to make sense of a mathematical situa-
tion. A mental mechanism is the means by which a mental structure is
constructed in the mind of an individual.
In APOS Theory, the mental structures are Actions, Processes, Objects, and
Schema. These structures are constructed through mental mechanisms such
as interiorization, coordination, encapsulation, and thematization. Specifically,
an Action is interiorized into a mental Process, two mental Processes can be
coordinated to form a new Process, a Process is encapsulated to form a mental
Object, and a Schema can be thematized into a mental Object.
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_10,© Springer Science+Business Media New York 2014
175
See Chap. 3 for more on mental structures and the mechanisms by which they
are constructed.
Q: In the mental development of a mathematical concept, must the learner run
through every stage of APOS, and if so, doesn’t that necessarily mean that each
stage must be constructed (or passed through) linearly?
A: Since a Process involves the reconstruction of an Action (via interiorization) and
since an Object arises (via encapsulation) out of the desire (or need) to apply an
Action or Process to a Process, full development of a mathematical concept
necessitates the construction of each stage.
This seems to suggest that the development of mental constructions always
proceeds linearly. However, this may not always be the case, particularly
when the learner is asked to apply a concept to an unfamiliar situation.
Although the learner first tries to assimilate the new situation, that is, to use
existing structures to make sense of the situation, this may not be possible.
When this is the case, the existing structures need to be reconstructed, that is,
the learner needs to accommodate the existing structures in order to assimilate
the new learning situation. The reconstruction typically involves a nonlinear
progression through the Action—Process—Object sequence. For example, a
learner may have previously constructed the concept of binary operation as a
mental Object. When presented with a new situation, say a function defined on
a set with which the learner is unfamiliar, he or she would need to
de-encapsulate the binary operation Object back to its underlying Process
and reconstruct her or his Process conception in order to assimilate the new
context.
Q: Can a learner encapsulate the “wrong” Process?
A: In general, any mental Process can be encapsulated. For instance, in the mental
construction of infinite repeating decimals, it is possible for a learner who has
constructed a repeating decimal as an infinite Process to encapsulate a finite
Process. This occurs when the learner does not yet conceive of an infinite
repeating decimal Process as a Totality. In an effort to apply an Action
(or Actions) to a repeating decimal (or repeating decimals), the learner may
encapsulate a finite Process. For the repeating decimal0:�9, a learner who does notsee the infinite decimal in Totality might see the decimal as infinitesimally close
to but not equal to 1 (see Chaps. 5 and 8 for more on the mental construction of
infinite repeating decimals).
Q: Often it is said that Actions or Processes can be applied to mental Objects. Can
you give an example of a Process applied to an Object? How does this differ
from an Action applied to an Object?
A: A Process is an Action which has been interiorized and is under the learner’s
control. As such, its steps do not need to be carried out explicitly. Thus, any
interiorized Action that a learner applies to an Object is necessarily a Process
applied to the Object. For example, given propositions A and B, where both are
conceived as Objects, an individual might apply an implication, that is, if A is
176 10 Frequently Asked Questions
true, then B is true. This thought does not require knowing whether A or B or
both are true, so it is not being done explicitly following any algorithm. By
definition, this is a Process.
Another example might involve operations on cosets. If the learner is given a
set of cosets and asked in general how to define an operation on a coset, the
operation would be conceived as a Process if no explicit expression for carrying
out the operation is given.
Q: It is sometimes difficult to apply APOS Theory in nonfunction-related contexts.
How can one distinguish between a Process and a function?
A: In a sense, since both Actions and Processes are transformations of Objects just
about everything to which APOS Theory is applied can be considered to be a
function. But each concept has its own features that must be considered explic-
itly, so thinking of everything only as a function may not be useful. Also, while
mathematicians do not always distinguish between a Process and a function (out
of convenience), one must be aware of the distinction (i.e., a Process is only one
part of a function) and maintain it where appropriate.
It is also important to note that APOS Theory has been applied successfully to
a variety of concepts that do not explicitly involve the concept of function (see
Chaps. 4, 5, 7–9, as well as Weller et al. 2003).
Q: Where can one learn more about the relation between APOS Theory and Piaget?
A: APOS is an extension of Piaget’s theory of reflective abstraction applied to
advanced mathematical thinking. If one accepts the idea that mathematics is the
study of mental objects and how they are transformed, APOS provides a
language and methodology that can be used to describe how individuals con-
struct and transform such objects, the mechanisms by which they are
constructed, and the role of instruction in helping students to make those
constructions. The relationship between Piaget’s theory of reflective abstraction
and APOS Theory is discussed in detail in Chap. 2.
Q: Does APOS Theory take into account Actions applied to physical objects?
A: This is discussed in Chap. 9.
10.2 Questions Related to Genetic Decomposition
Q: Is it possible that different genetic decompositions lead to different
understandings of the same concept?
A: A genetic decomposition is a description of the mental structures an individual
may need to construct in coming to understand a mathematical concept.
As discussed in Chap. 4, a genetic decomposition for a concept may not be
unique, that is, there may be multiple paths by which individuals construct their
understandings. The issue is whether a particular path can be verified empiri-
cally. If student data show evidence of mental constructions that align with a
10.2 Questions Related to Genetic Decomposition 177
particular genetic decomposition, then that description is validated. As pointed
out in Chap. 4, APOS-based research has not found very many examples of
different genetic decompositions for a single concept except in the case of
preliminary genetic decompositions.
Q: Reference to a genetic decomposition means that the mathematical object of
interest can be decomposed, but it is impossible to decompose many interesting
mathematical objects. How can it be possible that a cognitive process can be
decomposed in stages?
A: In APOS Theory, it is cognitive concepts and not Processes that are decomposed.
A genetic decomposition is a model that describes the mental structures that an
individual may need to construct in order to learn a mathematical concept. This
means that a genetic decomposition is not about decomposing an Object or a
Process. Rather, according to the theory, the construction of cognitive Objects
begins with Actions applied to known physical or mental Objects. As an
individual reflects on these Actions, a new type of construction develops, as
the Action is transformed into a mental Process. Reflection on the Process and
the need to perform Actions on the Process result in the encapsulation of the
Process into a cognitive Object. Thus, Actions, Processes, and Objects are not
parts into which a mathematical object is decomposed.
In the case of Schema, construction of relations among different Actions,
Processes, Objects, and Schemas makes the construction of a new Schema
possible, that is, a Schema is developed in terms of relations among its
components. While it is true that one can think of a Schema as composed of
these elements and can think of decomposing the Schema into its components,
the relations among the components are as important as the components
themselves.
10.3 Questions About Instruction and Performance
Q: Can APOS Theory be applied everywhere? Does it work for every topic and
concept?
A: APOS-based instruction has been designed and implemented for a wide variety
of concepts in the undergraduate curriculum. APOS Theory has been used in the
design of instruction and study of student thinking in the areas of mathematical
induction, quantification, calculus, functions, linear algebra, abstract algebra,
mathematical infinity, and repeating decimals. It has also been applied at the
K–12 level with fractions and algebraic thinking. So far, the theory has proven to
be effective for all of the concepts to which it has been applied (Weller et al.
2003).
Since mathematics involves the study of mental Objects and since the study of
mental objects involves transforming them, it is conceivable that APOS Theory
178 10 Frequently Asked Questions
can be applied to the study of the cognition and instruction of any mathematical
concept.
Whether APOS can be applied to concepts outside of mathematics is an open
question. At this point, there are some efforts to see whether APOS can be
applied to questions of cognition and instruction for concepts in computer
science.
Q: For performance on tasks involving a given concept, does limitation to an Action
conception necessarily suggest difficulty whereas an Object conception implies
success?
A: Action, Process, and Object are mental structures; they do not represent levels of
mathematical performance. However, it is possible to conflate these ideas. A
learner who is limited to an Action conception would have difficulty with tasks
that call for a Process or an Object conception. For instance, a learner who is
limited to an Action conception of the function concept would have difficulty
composing two functions that are not given by explicit formulas. Similarly, a
student who is limited to a Process conception of the function concept would
have difficulty determining the supremum of a set of functions. So, limitation to
an Action conception means that the learner is confined to tasks requiring no
more than Actions whereas a Process conception enables the learner to work on a
wider variety of tasks.
Q: How do traditional and reform-oriented approaches to instruction differ from an
APOS-based approach to instruction?
A: A traditional approach to instruction, usually with a focus on lecture as the
principal instructional strategy, emphasizes the dissemination of information.
The instructor’s role is to organize ideas related to a concept and then to present
those ideas as clearly as possible.
A reform-oriented approach to instruction, with a focus on non-lecture
instructional strategies, emphasizes learning through engagement. The
instructor’s role is to design individual and collaborative activities that help
students learn about a concept through experience.
An APOS-based approach to instruction emphasizes the construction of
mental structures that may be needed in the learning of a concept. The role of
the instructor is to identify the mental structures that might be needed in learning
the concept and to design activities that help students make the proposed mental
constructions.
Elements of traditional and reform-oriented approaches may be used in the
design of APOS-based instructional activities. This typically involves use of the
ACE Teaching Cycle, which is discussed in detail in Chap. 5.
Q: Different APOS papers say, “In our work, we have used cooperative learning
and implemented mathematical concepts on the computer....” Since APOS-
based pedagogy is not necessarily tied to cooperative learning and program-
ming, can other pedagogies be applied in the implementation of APOS Theory?
10.3 Questions About Instruction and Performance 179
A: Other pedagogies have been tried, for example, for elementary school students
(see Chap. 9), but it seems that for students at the collegiate level, the best results
are obtained when cooperative learning, along with writing and running
programs to represent Processes and Objects, is used.
10.4 Questions Related to Topics Discussed inMathematics
Education: Representations, Epistemology,
Metacognition, Metaphors, Context
Q: What is the role of APOS Theory in metacognition and reflection?
A: The underlying concept in APOS Theory is Piaget’s notion of reflective abstrac-
tion (Dubinsky 1991). According to Piaget, reflective abstraction has two
aspects. One is the reflection (and possible reconstruction) of a concept onto a
higher plane of thought. The other is reflection on an individual’s thinking about
a concept. Reflection and relations to metacognition are contained in the second
aspect.
Q: Does APOS Theory take into account representations of mathematical concepts?
If so, how would this occur?
A: Most of the mathematics education literature on representations concerns issues
involving the transition from one representation to another. APOS Theory
considers representations and transitions among them in a somewhat unique
manner (as illustrated in the accompanying figure for the concept of function).
Figure 10.1 shows an apex with several downward pointing arrows. The apex
represents the genetic decomposition. The arrows from the apex point down to
different representations. The key idea is that the student constructs the concept via
the genetic decomposition. In dealing with a problem situation, which may call for
a particular representation of the concept, the learner thinks of the concept in terms
of that representation. An arrow from the apex to one of the representations of the
concept accounts for this. If a student needs to change representations, that is, to
transfer from representation A to representation B, he or she moves from represen-
tationA (in her or his thinking) to the apex and then from the apex to representation
B. In the example of the function process illustrated below, the student uses the
Fig. 10.1 Transitions among function representations
180 10 Frequently Asked Questions
given representation to figure out the Process of the function that is represented.
Then, using her or his Process conception, the individual moves down (most likely,
in an unconscious way) along the line corresponding to the desired new represen-
tation to express the process in terms of the new representation. Instruction based
on this idea has been used in a high school class with promising results (Dubinsky
and Wilson 2013).
According to APOS Theory, the reason students have so much trouble
making the transition from one representation to the next is that they (are taught
to) go directly from one representation to another without passing through the
cognitive meaning of the concept (given by the genetic decomposition). Con-
siderably more research needs to be conducted to determine whether the point
of view suggested by APOS Theory is useful.
Q: Does APOS Theory take into account the epistemology of mathematical concepts?
A: The answer is yes. In fact, a genetic decomposition is an epistemological
analysis, in line with the genetic epistemology of Piaget. In it the nature of a
mathematical concept appears through mental structures and mechanisms that
might give rise to its construction.
Q: In APOS Theory, what does it mean to construct an understanding of a
mathematical concept?
A: APOS Theory considers that mathematical concepts are the building blocks of
mathematics. From the point of view of APOS Theory, constructing an under-
standing of a mathematical concept means that students are capable of dealing
with certain types of familiar and unfamiliar problem situations involving the
concept. According to the types of situations a person can deal with, an analysis
using APOS Theory describes that understanding in terms of mental structures
and their relationships. However, construction of a particular mental structure
does not necessarily mean that a student will deal successfully with a problem-
solving situation where the structure is needed. Rather, construction of a
particular structure would suggest that the student has the capability to deal
with problem-solving situations in which construction of the structure is called
for. A person who demonstrates a deep understanding of a concept is capable of
dealing with unfamiliar and even new situations using the concept or concepts
in question. See Chaps. 5, 6, 8, and 9 for a discussion related to this issue.
Q: APOS studies do not appear to pay particular attention to the contextual
interaction of subjects with mathematical concepts. Why?
A: APOS Theory deals with the construction of mental structures that may be
needed in the development of mathematical knowledge. The theory implicitly
assumes that individuals need to construct certain mental structures before they
can deal with mathematical contexts involving the concept. The importance of
context in the learning of mathematical concepts is worthy of further inquiry,
although there seems to be little evidence that studying mathematics in context
improves learning and even some indication that it might be counterproductive
in some cases.
10.4 Questions Related to Topics Discussed in Mathematics Education. . . 181
Q: Does APOS Theory take into account the construction of metaphors or
Grundvorstellungen (simple ideas)?
A: Simple ideas, or Grundvorstellungen, are taken into account in APOS Theory in
the sense that the construction of new cognitive Objects starts from previously
constructed Objects that may be based on simpler ideas. Metaphors are not taken
into account in APOS Theory, which is an alternative to metaphors as a means of
describing the construction of new knowledge.
10.5 A Question About Intuition
Q: What role does intuition play in APOS Theory?
A: According to Piaget, intuitions, like all thought, are constructed (Beth and Piaget
1965/1974). Generally speaking, the usefulness of an intuition depends on its
effect on a student’s ability to make the specific mental constructions called for
by a theoretical analysis. In APOS Theory, instruction for a concept is based on
its genetic decomposition, a description of the mental constructions a student
needs to make in order to learn the concept. According to the theory, a student
can develop understanding of any concept for which he or she has made the
necessary mental constructions. Within this context, there are two types of
intuitions—those that support development of the proposed mental
constructions or those that impede its development.
For example, when comparing the cardinality of infinite sets, students some-
times try to extend to infinite sets the part–whole notion developed in comparing
finite sets. In working with finite sets, students see that the cardinality of Bexceeds the cardinality of A if A is a proper subset of B. In seeing that this
relationship holds for every pair of finite sets A and B (for which A is a proper
subset of B), a student constructs an intuitive scheme, that is, the student
naturally and unconsciously thinks of and applies this relationship when com-
paring the cardinality of finite sets. When presented with two infinite sets A and
B, with A a proper subset of B, the learner may apply the constructed intuition,
believing that a part–whole comparison, which has worked successfully for
finite sets, applies to infinite sets. A familiar case involves comparison of the
even counting numbers and the natural numbers. Many students believe the
former has a smaller cardinality than the latter because the even numbers are a
proper subset of the natural numbers. In a situation such as this, the role of
APOS Theory is not to dismiss the usefulness of the intuition but to enhance
accommodation of the existing structure of the intuition, so it treats infinite sets
differently. Large finite sets and infinite sets have a common characteristic:
neither can be physically enumerated. The two sets differ, though, in how one
imagines their enumeration—a large finite set has a last element while an infinite
set does not. For a large finite set, the last element signifies completion; the two
sets can be counted and the results of each count compared. For an infinite set,
since there is no last element, completion is not indicated by a last element but
182 10 Frequently Asked Questions
by completion of the Process, that is, by the ability to conceive of the Process ofenumeration in the past tense. As a result, the usual notion of counting does not
apply. The difference in the meaning of completion between finite and infinite
sets explains why part–whole, which works for finite sets, fails for comparison
of infinite sets and why a different means of comparison is called for.
In both cases, comparison is an Action applied to sets. For this Action to be
applied, the Process must be encapsulated. For finite sets, this is triggered by
enumeration of the last element. For infinite sets, things are more complicated.
Specifically, the completed Process of enumeration must be viewed in its
totality, that is, as a single operation freed from temporal constraints. This
transition from Process to Object is what makes the comparison of infinite
sets so difficult and is what helps to explain the persistence of part–whole
thinking, even when it does not apply.
On the other hand, intuitions can prove to be useful, even when crossing
domains such as the transition from finite to infinite. For a finite sequence of
iterated actions in construction of a set (e.g., a nested sequence of sets), one
would conclude that an element that arises at step n and for all successive steps
beyond step n would be contained in the final set constructed by the iteration
(here called the “principle of accumulation”). This principle guides the iterative
construction of any finite set (e.g., the first k counting numbers) and also applies
to construction of the set of natural numbers N: once a natural number appears,
as one iterates, it appears in the set constructed at each subsequent step and in
the resulting set, which is N itself. This construction is rooted in one of the most
elemental mathematical activities—counting—which is the basis for any pro-
cess of incremental accumulation. Thus, it is reasonable to say that individuals
likely construct the principle of accumulation as an intuitive notion.
Radu and Weber (2011) provide some confirmation for this in their study of
students’ thinking about completed infinite iterative processes. On one task,
students are asked to determine the state at infinity, or resultant state, for the
Vector Problem, which follows:
Let v ¼ ð1; 0; 0; . . .Þ 2 NN .This vector will be modified in the following ways:
Step 1: v ¼ ð0; 1; 2; 0; 0ÞStep 2: v ¼ ð0; 0; 1; 2; 3; 0; 0ÞStep 3: v ¼ ð0; 0; 0; 1; 2; 3; 4; 0; 0ÞIf this process is continued ad infinitum, what form does v take after all of the steps have
been completed? (p. 167)
Without prior instruction, the students solved the problem correctly, explained
why their solutions made sense, and used their work to solve other related infinity
tasks. Radu and Weber attributed the students’ success, in part, to their construc-
tion of correct intuitions, in particular, to the principle of accumulation.
The students did not perform as well on a similar problem, referred to here as
the Tennis Ball Problem:
Suppose that an infinite set of numbered tennis balls and a large table are available. Place
balls numbered 1 and 2 on the table and remove number 1. Next, place balls 3 and 4 on
10.5 A Question About Intuition 183
the table and remove number 2. Then place balls 5 and 6 on the table and remove number
3.And so on, ad infinitum. What happens after all of the steps have been carried out?
(Radu & Weber 2011, p. 172)
The difference between the Vector Problem and the Tennis Ball Problem lay
in the context of each problem. According to Radu and Weber, the Vector
Problem focuses attention exclusively on the elements and positions of the
natural numbers (in the representation of each vector) whereas the Tennis Ball
Problem includes cardinality (the number of balls that remain on the table at
each step increases by one). The issue of cardinality is what makes the Tennis
Ball Problem paradoxical: it seems that one cannot determine how many balls
are on the table at the end of the activity, because, on the one hand, the number
increases by one at each step, which implies an infinite quantity, but, on the
other hand, given any tennis ball, one can say exactly when that ball is removed
so that none is left. The latter explanation is correct: since ball n is removed
from the table at step n and remains removed for every subsequent step, it
follows that every ball is removed.
APOS Theory can be used to explain why the principle of accumulation
supports student thinking in the Vector Problem and why the paradox seems to
blunt its effect in the Tennis Ball Problem. According to Brown et al. (2010), an
iterative process is based on understanding iteration through N: the process,
which begins at 1 and at each successive step adds 1, results in the sequence 1, 2,
3, . . ., which leads to the construction of sets: {1}, {1,2}, {1,2,3}, . . .. Embed-
ded in this construction is the idea that once a natural number appears, it appears
for every subsequent step. Since the construction is cumulative, encapsulation
produces a final object that includes every natural number. Thus, the principle of
accumulation supports the mental construction of the set. The same idea stands
behind the construction of the infinite zero vector in the Vector Problem—the
number 0 is added at step n in position n and appears in every subsequent vectorconstructed at every subsequent step. The accumulation principle then leads one
to conclude that the state at infinity consists of the infinite zero vector.
On the other hand, the issue of cardinality confounds one’s ability to solve the
Tennis Ball Problem. Since the number of balls on the table increases by one at
each step, one is tempted to conclude that there are infinitely many balls that
remain on the table when the procedure has been fully carried out. This view is
plausible if the iterative process of placing and removing balls is viewed as
incomplete. Determination of the result of the procedure is an Action applied to
the iterative Process. If that Process is not seen as complete, the individual
encapsulates what is, for all practical purposes, a finite Process. As a result, he or
she concludes that balls remain on the table. To move beyond that, the individ-
ual needs to see the infinite process as a completed totality. This enables her or
him to see that the relevant correspondence, the removal of ball n at step n, is asingle operation applied to each ball. Although the Vector Problem and the
Tennis Ball Problem are similar mathematically (they both involve a movement
of natural numbers that leads to a cumulative result), the Tennis Ball Problem
184 10 Frequently Asked Questions
represents that movement in a way that makes it more difficult to solve. This is
consistent with what Tirosh and Tsamir (1996) found in their study of infinite
comparison tasks—the representation of the task strongly influences students’
reasoning about the task. The APOS analysis provides a theoretical explanation
for why the Tennis Ball Problem is difficult to solve and shows why the intuitive
principle of accumulation supports the mental construction of the natural num-
bers N, which makes the Vector Problem more readily solvable.
Finally, an individual’s Schema for different infinity concepts necessarily
includes different intuitions regarding the concept. The role of what in APOS
Theory is called coherence of the Schema provides the mechanism by which an
individual decides which intuition to use in given problem situations. Whether a
particular intuition is useful depends on whether it supports the other mental
structures that constitute the Schema as well as the relationship among those
structures.
10.6 Questions About How Specific Concepts
Can Be Approached with APOS Theory
Q: In performing Actions on Objects, some researchers wonder why many learners
do not connect activities with concrete manipulatives with formal operations.
This question is often raised in relation to fractions (see Freudenthal 1973;
Herman et al. 2004; and Chap. 9). Can APOS Theory be used to explain why
this connection may or may not take place?
A: APOS Theory claims that such a connection does not develop automatically but
as a result of an appropriate choice of manipulatives and learning sequences (see
Chap. 5 for a detailed discussion of APOS-based instruction) that enhance the
development of a specific concept through the stages Action–Process–Object.
Without evidence of these two—appropriate manipulatives and adequate
learning sequences—the connection between manipulatives and formal
activities cannot be assumed.
Q: For a given set S, a binary operation o defined on S, and an operation problem
a o b ¼ c for a; b; c 2 S , is the element c , the result of the operation, the
object that results from encapsulation of the process of applying o to a; b 2 S?A: The Object obtained from the encapsulation of a binary operation is not the same
as the Object that results from the application of that binary operation to two
specific elements of the set. For example,
[I]n the addition of two numbers, say 2 and 6, the number 8 is obtained from the process of
adding 2 and 6 but is not the object that results from encapsulation of the binary operation
process. Instead, the encapsulation allows the addition to be considered as an object that can
be acted on; for example, it could be compared to other processes like 6 + 2 and 8 – 6. The
number 8, like other natural numbers, is an object that is constructed by encapsulating
processes other than the basic arithmetic operations (Piaget, 1952 [sic][1941/1965]).
(Dubinsky et al. 2005b, p. 260)
10.6 Questions About How Specific Concepts Can Be Approached with APOS Theory 185
If the number 8 had not been previously constructed, that is, by encapsulating
the Processes that would lead to its construction, then the individual would
probably be unsuccessful in performing arithmetic operations to obtain 8 as a
result.
Q: In studying proportions, fractions ab and non-numeric ratios such as “a is to b”
arise. A possible APOS interpretation is that such a relation is a transformation
from a to b which could be an Action or a Process. The question is how does an
observer decide which it is?
A: One response is that it is not easy to tell what the subject is able to do in working
with the situation. As a result, one must conduct an interview with the subject to
see how he or she is thinking about the situation.
Another, perhaps better, response is that if the subject is unable to reverse the
relation or coordinate it with other relations, then it is likely that the subject has
no more than an Action conception.
Q: Many students are confused about the difference between a set A being an
element of a set B and the set A being a subset of the set B. How is this explained
by APOS Theory?
A: One needs to establish the notion of a set as an Action and later, as a Process, of
placing objects into a container. This can be done initially using physical
containers or bags of physical objects. Then A is an element of B if the entire
container (not just its contents) is contained in B. The meaning of A being a
subset of B is that every time an individual selects an element of A and tries to
determine whether it is contained in B, the individual finds that it is already
contained in B. Activities involving objects and containers can help learners to
make this distinction. Whether these types of activities would result in construc-
tion of the proposed mental constructions is a subject for future research.
Q: How can APOS Theory be used to answer the previous question for the case in
which A is the empty set?
A: The response is the same as for the previous question. The empty set is
represented by an empty container.
Q: Many subjects have difficulty accepting the fact that the empty set is a subset of
every set. How is this dealt with using APOS Theory?
A: One way that helps with such issues is to always (at least until fundamental ideas
about sets are well established) have a universal set explicitly involved when
talking about sets. Then the empty set is the complement of the universal set.
Another approach is to make use of the notion of a set as an Action and later as
a Process of placing objects into a container. Then, as discussed in the previous
two questions, the meaning of A as a subset of B is that every time an individual
selects an element of A and tries to put it in B, he or she finds that it is already
contained in B. If A is the empty set, this condition is always satisfied, but
vacuously, which may make it harder to understand. Activities in which one
186 10 Frequently Asked Questions
tests for a subset by emptying the contents of bag A into a bag B to see that nothing
in B has changed can help. The condition is always satisfied if A is empty.
Q: APOS studies of the learning of cosets do not appear to take into account
geometric representations of cosets, that is, as objects that are points in a
geometric space. Would the consideration of this type of representation require
the need for a different genetic decomposition of the coset concept?
A: There do not seem to be any studies that consider cosets as points in a geometric
space. Whether construction of this representation would be described by a
different genetic decomposition is a topic for further research.
10.6 Questions About How Specific Concepts Can Be Approached with APOS Theory 187
Chapter 11
Conclusions
The discussion concludes with three themes that have been developed throughout
the preceding chapters. The notions of the developmental/evaluative dichotomy, the
development of APOS Theory through mechanisms similar to those used in the
theory, and the future of APOS Theory are discussed. The chapter provides “APOS
Theory at a glance” and ends with some final thoughts.
11.1 Developmental vs. Evaluative Nature
The research question of how a person may learn a particular concept (or topic or
subject area) is readily seen as too complex to be able to address in a single study.
The variables at work are too wide ranging and sometimes cannot be directly
observed and measured. In response, researchers must choose a lens, or a filter,
through which to explore how learning takes place. Theories of learning and
associated frameworks serve as guides for the researcher to explore more finely
focused aspects of the research question. Some frameworks analyze the learning
environment, pedagogical strategies, or curricular materials to be employed. Others
explore student–student interactions or student–instructor interactions in problem
situations. A third type of framework analyzes the types of mental constructions
that may be made by an individual as learning takes place.
APOS Theory primarily falls into this third type, cognitive studies. It proposes
certain mental structures that may be necessary for the construction of a concept
along with the mechanisms for building those structures. In this aspect, APOS
Theory serves as an evaluative framework as individuals are observed in problem
situations in which the researcher attempts to describe their level of understanding
as well as the mental structures at work in their learning of the concept. Chapter 2
described in detail the connections to Piaget’s work and gave the background for
using APOS Theory to analyze how an individual might learn a concept. Chapters 3
and 7 elaborated on the structures and mechanisms used by APOS Theory to
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_11,© Springer Science+Business Media New York 2014
189
evaluate a particular individual’s state of learning. Chapter 4 explained that a
genetic decomposition describes what a general learner might need to construct
the concept under study. The genetic decomposition becomes the working hypoth-
esis that is used to evaluate the degree to which learning has taken place.
APOS Theory also may be considered to be a developmental framework for the
design and implementation of instructional materials and settings. These pedagogi-
cal strategies are typically constructed based on a genetic decomposition (prelimi-
nary or revised). They use problem situations often based on computer activities
that help the students consider new mathematics or nuances in previously seen
ideas. The ACE Teaching Cycle used by APOS Theory was discussed in Chap. 5. In
Chap. 6, it was seen how the research methodology is used to assess the effective-
ness of APOS-based pedagogy as well as to evaluate the theoretical description
used in its development.
11.2 Macro-Level Consistency
Reflective abstraction is the mechanism that drives construction of new mental
structures in APOS Theory. As discussed in Chap. 3, generalization allows a learner
to construct a new Process from existing Processes. Both assimilation and accom-
modation are examples of generalization. With assimilation, new situations are
dealt with by using existing structures in a new way. Accommodation involves the
reconstruction of an existing structure in order to deal with an unfamiliar situation.
The evolution of APOS Theory as discussed in Chap. 2 and observed in Chaps. 8
and 9 seems to have progressed by means of similar mechanisms. Notable progress
was found in the study of limits (Cottrill et al. 1996), the chain rule project (Clark
et al. 1997), and the work with graphing via derivatives (Baker et al. 2000; Cooley
et al. 2007). The obstacle that the researchers found in the limit study and in the
chain rule study was an inability to identify the Processes that might be
encapsulated into an Object conception. The solution in the limit study was to
recognize that two Processes are coordinated. This was an assimilation of our
existent understanding of APOS Theory. A reconstruction was necessary in the
case of the chain rule where the limitations of the theory caused an in-depth
exploration of what a Schema is and how it might be described. Thus, APOS
Theory was accommodated in order to assimilate the triad of stages found in the
work of Piaget and Garcia (1983/1989) as discussed in Chap. 7.
The projects that studied how students come to learn to apply derivative infor-
mation to construct the graph of a function found the triad descriptions lacking in
the attempt to evaluate the observations. The first study analyzed the interaction
between Schemas, extending the theory’s use of the triad stages (Baker et al. 2000).
In the second paper (Cooley et al. 2007), the researchers then found it necessary to
unpack the triad descriptors and reconstruct them along with the mechanism of
thematization, which is described in Chap. 7. This sort of accommodation to APOS
190 11 Conclusions
Theory was also seen in the introduction of Totality in the study of infinite repeating
decimals (Dubinsky et al. 2013) described in Chap. 8.
Thus, the fundamental impetus for modifying APOS Theory is also reflective
abstraction. As a hypothesis (genetic decomposition) is tested, the data may (1)
support the proposed constructions, (2) offer an obstacle that presses the theoretical
description and requires an assimilation by reworking the description, or (3) require
a reconsideration of the theory underlying the description that results possibly in an
accommodation of the theory to account for the data.
11.3 A View of the Future of APOS
As seen in Chaps. 8 and 9, APOS Theory continues to evolve as a potential new
structure (Totality) is investigated and is being applied to other areas of mathemat-
ics (as it moves into elementary mathematics). Analyses of students’ understanding
of infinity suggest the need for a mental structure apart from Process and Object.
These studies, described in Chap. 8, also suggest the need for levels between the
stages of Action and Process, between Process and Totality, and between Totality
and Object. Studies will need to be designed and implemented to clarify whether
this new stage exists in topics other than that of infinite repeating decimals.
In Chap. 9, the work of Arnon and her colleagues on students learning elemen-
tary mathematics was described. The investigations deal with concrete objects in
order to begin the development of mental Objects. The research into students’
understanding of fraction also explores the notions of levels between stages. Their
work involves students at the stage of concrete operations prior to moving to formal
operations. APOS Theory might be used to explore learning of other elementary
mathematical topics.
There remain many topics in undergraduate mathematics that have not been
investigated via the lens of APOS Theory. Other topics have been studied and the
results published before our current understanding of thematization of Schemas.
One example is the paper on limits (Cottrill et al. 1996), which has served as a
useful example of our research paradigm (discussed in Chap. 6). It would be
beneficial to reconsider the topic of the limit of a function at a point from the
point of view of constructing a Limit Schema, which may include other types of
limits as well.
11.4 APOS Theory at a Glance
APOS Theory is based on the following principle:
An individual’s mathematical knowledge is her or his tendency to respond to perceived
mathematical problem situations by reflecting on problems and their solutions in a social
context and by constructing or reconstructing mathematical actions, processes and objects
and organizing these in schemas to use in dealing with the situations. (Asiala et al. 1996, p. 7)
11.4 APOS Theory at a Glance 191
This statement was made by Dubinsky early in the work on APOS Theory and
normalized for the article mentioned.
11.4.1 Structures and Mechanisms
Three basic types of knowledge—Actions, Processes, and Objects—are involved in
mathematical concept construction and are organized into structures called
Schemas. An Action is any repeatable physical or mental manipulation of Objects
to obtain other Objects. It is a transformation that is a reaction to stimuli that the
individual perceives as external.
As an individual reflects on an Action, it is interiorized and becomes a Process.
With a Process conception, the learner perceives the Action as part of her or him
and has control over it. As the individual realizes that an Action can be brought to
operate on a Process, the Process is encapsulated to become an Object. The Object
can be de-encapsulated back to the Process as needed. Processes may also be
constructed by reversal and coordination mechanisms.
A Schema is a coherent1 collection of Actions, Processes, Objects, and other
Schemas that is invoked to deal with a new mathematical problem situation. A
Schema can be thematized to become another kind of cognitive Object to which
Actions and Processes can be applied. By consciously unpacking a Schema, it is
possible to obtain the original Processes, Objects, and other Schemas from which
the Schema was constructed (Chap. 7).
The triad, introduced by Piaget and Garcia (1983/1989), distinguishes three
stages in the development of a Schema: Intra-, Inter-, and Trans-. In APOS Theory,
the Intra-stage is characterized by a focus on a single Object in isolation from any
other Actions, Processes, or Objects. The Inter-stage is characterized by
recognizing relationships between different Actions, Processes, Objects, and/or
Schemas. It is useful to call a collection at the Inter-stage of development a
pre-schema. Finally, the Trans-stage is characterized by the construction of an
overall structure underlying the relationships discovered in the Inter-stage of
development.
It is worth noting that it is only when a schema reaches the Trans-stage of
development that it can properly be referred to as a Schema in APOS Theory. The
reason is that at the Trans-stage, the underlying structure is constructed through
reflecting on the relationships among the various Actions, Processes, Objects, and/
or Schemas from the earlier stages. This structure provides the necessary coherence
in order to identify the collection as a Schema, that is, as a coherent whole. This
coherence consists in deciding what is in the scope of the Schema and what is not.
1 The definition of coherent is found two paragraphs below.
192 11 Conclusions
11.4.2 Research Methodology
A genetic decomposition is a hypothetical model that describes the mental
structures and mechanisms that a student might need to construct in order to learn
a specific mathematical concept. It typically starts as a hypothesis, called a prelimi-
nary genetic decomposition, based on the researchers’ experiences in the learning
and teaching of the concept, their knowledge of APOS Theory, their mathematical
knowledge, previously published research on the concept, and the historical devel-
opment of the concept (see Sect. 4.1).
A genetic decomposition might be used in two ways: (1) to develop pedagogical
materials and settings or (2) as the hypothesis for collecting data to test the model.
In the first case, instruction takes place followed by data collection. The data are
then explored to determine how learning has taken place—evaluating the instruc-
tional approach—as well as testing the genetic decomposition. In the second case,
the genetic decomposition acts as the research hypothesis to be tested by empirical
data (see Chap. 6).
Data are collected via instruments that may include written questionnaires, semi-
structured interviews (audio- and/or videotaped), exams, and/or computer games.
All written work from questionnaires and interview tasks are collected. Triangula-
tion of observations is enhanced by collaborative research as each researcher
negotiates her or his analysis with the others. Chapter 6 describes a process for
scripting interview transcripts and analyzing trends that emerge.
11.4.3 Pedagogical Approach
The design and implementation of instruction is one of the three components of the
research paradigm of APOS Theory. This component involves the ACE Teaching
Cycle and cooperative learning techniques. These provide the “perceived mathe-
matical problem situations” and “social context” mentioned in the statement of the
principle in the beginning of this section.
The ACE cycle begins with activities (typically involving computer program-
ming experiences) that ask the students to consider problems that lie just beyond
their shared experience. The activities may be extensions of previous experiences or
situations that lead the students through an algorithm. The classroom discussion is
based on the shared experience of the activities. The discussion allows students to
analyze the experience together and reflect on the important aspects of the activity
set. The third component of the cycle consists of homework exercises that continue
the reflections on the activities and discussion as well as extend the students’
experiences with the topic at hand.
The students work cooperatively on the activities in order that mathematical
notions may be learned through discussion. As the students work in groups, the
activities may involve a more complex situation than might be reasonable for an
11.4 APOS Theory at a Glance 193
individual to handle. Alternatively, the activity may be ambiguously stated so that a
negotiation of its meaning leads to the necessary reflection on the mathematics
being studied.
11.4.4 An Integrated Theory
APOS Theory provides descriptions of the mental structures and mechanisms that
may be necessary to describe a student’s apparent learning of a concept. With such
tools, it is possible to build a genetic decomposition of a concept for a generic
student. Using the genetic decomposition to design instruction based on the ACE
cycle enhances the value of the model. One might employ cooperative learning
techniques and activities in the classroom, but without a guiding model, these may
not lead to the type of learning that is desired. The research component provides the
empirical evidence to test the validity of the model and the efficacy of the instruc-
tion. The analysis is based on the genetic decomposition and at the same time
informs it. As the model is refined, the activities, discussion, and exercises are
modified and improved. Thus, all three components of the research and curriculum
development cycle—theoretical analysis, design and implementation of instruction,
and collection and analysis of data (Sect. 6.1)—are essential to APOS Theory.
11.5 Last Word
The intent of this book is to better explain issues that sometimes are not clear for
readers or students who try to understand or use APOS Theory. In particular,
Chaps. 2, 4, and 7 collect and synthesize ideas that had been spread among many
prior publications. Chapter 2 gives as complete a history as possible of APOS
Theory, written, as is the entire book, by those who have lived it. It is, of necessity,
incomplete because APOS Theory still lives and its story continues with little sign
of an ending.
Chapter 4 stresses that a genetic decomposition is a predictive model that can be
refuted or supported by experimental data. Data can also be used in the refinement
of a genetic decomposition; this possibility gives empirical support for the model.
The design of a genetic decomposition is one of the most difficult aspects of
applying APOS Theory to research on students’ understanding of mathematical
concepts and to teaching.
Chapter 7 demonstrates how the investigations into the notion of Schema show
consistently that even though students can use specific concepts to solve even
difficult mathematical problems, their understanding consists (possibly for a long
time) of separate domains and that the development of relations and transformations
between them is fundamental to achieving a deep understanding of mathematics.
These studies also show how Schema development mechanisms are useful tools to
194 11 Conclusions
understand students’ needs to develop richer Schemas to be able to develop the kinds
of tasks that may be needed to help them construct these richer Schemas that can
flexibly interact with other Schemas and can be thematized.
As shown throughout this book, APOS Theory, as a developmental and an
evaluative approach, can be useful to answer research questions about the learning
of a variety of mathematical concepts and to design and implement pedagogical
approaches so that learning occurs. As an active theory in continuous development
itself, the feedback it receives from research is reflected through new elements, and
this in turn gives rise to new research studies. The mathematics education commu-
nity, through the numerous publications listed in the annotated bibliography of this
book and many other works that are in development, has contributed to this growth
and, it is hoped, will continue to do so.
11.5 Last Word 195
Chapter 12
Annotated Bibliography
This chapter contains over 120 publications about APOS Theory. These
publications span a period of over 25 years and include research studies conducted
all over the world. The list of studies is not exhaustive but is representative. Some
publications that appear in this chapter might present perspectives on APOS Theory
that differ somewhat from those presented in this book. The reader is invited to
discern such differences, if found. The annotations that appear in this chapter either
were written by authors of the study or have been adapted from various sources, for
example, Dubinsky and McDonald (2001).
12.1 A Through B
One annotation appears for the following three studies:
Arnon, I. (1998). In the mind’s eye: How children develop mathematical concepts—ExtendingPiaget’s theory. Unpublished doctoral dissertation, School of Education, Haifa University.
Arnon, I., Nesher, P., & Nirenburg, R. (1999). What can be learnt about fractions only with
computers. In O. Zaslavsky (Ed.), Proceedings of the 23rd Conference of the InternationalGroup for the Psychology of Mathematics Education (Vol. 2, pp. 33–40). Haifa, Israel.
Arnon, I., Nesher, P., & Nirenburg, R. (2001). Where do fractions encounter their equivalents?
Can this encounter take place in elementary school? International Journal of Computers forMathematical Learning, 6, 167–214.
The studies by Arnon and her colleagues deal with the development of mathematical
concepts by elementary school children. The authors describe the difference between the
use of APOS at the postsecondary level and its use in elementary school. Using a
framework that combines APOS Theory with Nesher’s theory of Learning Systems, they
investigate the introduction of mathematical concepts as Actions on concrete Objectsversus their introduction as ready-made concrete representations. Yerushalmy’s ideas
(1991) of multiple representations were added to the above perspectives to develop
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6_12,© Springer Science+Business Media New York 2014
197
software adapted to the learning of the concept of equivalence classes of fractions. In these
studies, developmental paths for certain fraction concepts are established. It was found that
students who received instruction in which fractions were introduced as Actions on
concrete Objects progressed better along these paths than students who received instruction
in which fractions were introduced as ready-made concrete Objects. The findings also
established the following characteristic of students’ development of Actions on concrete
Objects into abstract Objects: after abandoning concrete materials, and before achieving
abstract levels, children perform the Actions in their imaginations. This corresponds to
interiorization (the passage from Action to Process) in APOS Theory.
Artigue, M. (1998). Ensenanza y aprendizaje del analisis elemental: ¿que se puede aprender de las
investigaciones didacticas y los cambios curriculares? Revista Latinoamericana deInvestigacion en Matiematica Educativa, 1(1), 40–55.
In the first part of this paper, the author discusses a number of student difficulties using
various theories of learning including APOS Theory. She explains that students are
generally unwilling to accept the equality 0.999. . . ¼1 because they see the former as a
Process and the latter as an Object. To accept the equality, both 0.999. . . and 1 must be
conceived as Objects. However, as Artigue points out, it is very difficult for students to
make the necessary encapsulation. In the second part of the paper, the author discusses the
measures that took place in France during the twentieth century to help students overcome
these difficulties.
Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A
framework for research and curriculum development in undergraduate mathematics education.
In Research in Collegiate mathematics education II. CBMS issues in mathematics education
(Vol. 6, pp. 1–32). Providence, RI: American Mathematical Society.
In this paper, the authors give a complete description of APOS Theory as it stood at the
time; a description of the ACE Teaching Cycle, the main pedagogical strategy for APOS-
based instruction; and the methodology used in APOS-based research. The combination of
the three has become known as “the APOS research framework.”
Asiala, M., Brown, A., Kleiman, J., & Mathews, D. (1998). The development of students’
understanding of permutations and symmetries. International Journal of MathematicalLearning, 3, 13–43.
The authors examine how abstract algebra students might develop their understandings of
permutations of a finite set and symmetries of a regular polygon. They give an initial
theoretical analysis of these topics, expressed in terms of APOS Theory, describe an
instructional approach designed to encourage development of the mental constructions
postulated by the theoretical analysis, and discuss the results of individual interviews and
performance on written examinations. The results indicate that the pedagogical approach
was reasonably effective in helping students develop strong conceptions of permutations
and symmetries. The authors also used the data to propose a revised epistemological
analysis of permutations and symmetries and to offer pedagogical suggestions.
Asiala, M., Cottrill, J., Dubinsky, E., & Schwingendorf, K. (1997a). The development of students’
graphical understanding of the derivative. Journal of Mathematical Behavior, 16, 399–431.
In this study, the authors explore calculus students’ graphical understanding of a function
and its derivative using APOS Theory. They present an initial theoretical analysis of the
cognitive constructions for development of the concept, outline an instructional treatment
designed to foster formation of the proposed mental constructions, discuss the results of
198 12 Annotated Bibliography
interviews conducted after the implementation of the instructional treatment, and describe a
revised epistemological analysis based on analysis of the data. Comparative data suggest
that students who received instruction based on the theoretical analysis were more success-
ful in developing a graphical understanding of a function and its derivative than students
who received traditional instruction.
Asiala, M., Dubinsky, E., Mathews, D., Morics, S., & Oktac, A. (1997b). Development of
students’ understanding of cosets, normality and quotient groups. Journal of MathematicalBehavior, 16, 241–309.
Using an initial epistemological analysis from Dubinsky et al. (1994), the authors deter-
mine the extent to which APOS Theory explains students’ mental constructions of the
concepts of cosets, normality, and quotient groups. They evaluate the effectiveness of
instructional treatments developed to foster students’ mental constructions and compare the
performance of students receiving this instructional treatment with those completing a
traditional course.
Asiala, M., & Dubinsky, E. (1999). Evaluation of research based on innovative pedagogy used inseveral mathematics courses. Unpublished report, available from the authors.
During three academic years from Fall 1997 through Spring 2000, APOS Theory was used
to teach a number of mathematics courses at Georgia State University. This study attempts
to assess the effectiveness of that approach in terms of improvement in students’ learning
and students’ attitudes toward mathematics. The manuscript contains an overview of the
literature that describes similar attempts by others at different universities. The results of
this study show an improvement, sometimes over time, of students’ attitudes and their
learning.
Ayers, T., Davis, G., Dubinsky, E., & Lewin, P. (1988). Computer experiences in the teaching of
composition of functions. Journal for Research in Mathematics Education, 19, 246–259.
Students from two sections of a college mathematics lab (n ¼ 13) who were given
computer experiences to encourage reflective abstraction scored higher on a test of their
understanding of functions and composition of functions than students from another section
(n ¼ 17) who were taught using traditional methods. The comparison was based on
questions intended to indicate whether reflective abstraction had taken place.
Badillo, E., Azcarate, C., & Font, V. (2011). Analisis de los niveles de comprension de los objetos
f0(a) y f0(x) en profesores de matematicas. Ensenanza de las Ciencias, 29(1), 191–206.
This paper describes the level of understanding of the relation between f0(a) (the derivativeof a function at a specified point) and f0(x) (the derivative at an unspecified point) among
five mathematics teachers who were teaching 16–18-year-olds in different schools in
Colombia. The analysis is based on APOS Theory with the addition of certain semiotic
aspects. The five teachers responded to an indirect questionnaire about their understanding
of f0(a) and f0(x) and were subsequently interviewed in relation to a series of vignettes.
Results illustrate how the comprehension of these two macro-objects, f0(a) and f0(x), can berelated to the structure of both graphic and algebraic schemes and the associated semiotic
conflicts.
Baker, B., Cooley, L., & Trigueros, M. (2000). A calculus graphing schema. Journal for Researchin Mathematics Education, 31, 557–578.
The authors used APOS Theory to analyze students’ understanding of a complex calculus
graphing problem that involved sketching the graph of a function on specific intervals of the
domain when given certain analytical properties. The data analysis uncovered the three-
12.1 A Through B 199
tiered development of Schema referred to as the triad of Schema development and two
Schemas that were interacting in the solution of the problem. One Schema involved
intervals and the second involved analytical properties. The authors also showed that the
interaction of these two different Schemas played an important role in the explanation of
many of students’ known difficulties.
Baker, B., Trigueros, M., & Hemenway, C. (2001). On transformations of functions. In
Proceedings of the Twenty-Third Annual Meeting, North American Chapter of the InternationalGroup for the Psychology of Mathematics Education (Vol. 1, pp. 91–98).
This study focuses on the analysis of student understanding of transformations.APOSTheory
was used as a theoretical framework to come up with a genetic decomposition for the concept
of transformation. The genetic decomposition was used to analyze class work and interviews
with 24 college students who had taken a precalculus course based on transformations of
functions. The course included writing and the use of graphing calculators. This paper
analyzes students’ difficulties related to the concept of transformation and the efficacy of
writing and calculators as teaching tools. Results showed that students tend to develop a
strong dependency on calculators to visualize functions, yet the use of calculators together
with writing assignments seemed to help with development of the transformation concept.
Results also suggest that this concept proved to be difficult for students.
Barbosa Alvarenga, K. (2003). La ensenanza de inecuaciones desde el punto de vista de la teorıa
APOE. Revista Latinoamericana de Investigacion en Matematica Educativa, 6(3), 199–219.
This study, which is based on APOS Theory, discusses mental constructions that under-
graduate students might make when trying to understand the inequality concept. This
involves many notions that must be coordinated: order of real numbers, factorization,
functions, function roots, 1–1 correspondence of real numbers with the number line,
equations, graphs, and graphical analysis of functions, implication, and equivalence.
Based on the construction of a Schema for inequality, the authors elaborate on a methodol-
ogy that will improve the teaching and learning of inequalities.
Baxter, N., Dubinsky, E., & Levin, G. (1988). Learning Discrete Mathematics with ISETL. NewYork: Springer.
This is the first textbook based entirely on the use of computer programming together with
APOS Theory. It was written before the ACE pedagogical structure was developed. The
subject matter includes topics for a college-level course in discrete mathematics: proposi-
tional and predicate calculus, sets and tuples, functions, combinatorics, matrices,
determinants, mathematical induction, and relations and graphs. For each concept, the
authors developed a genetic decomposition. The genetic decompositions guided the design
of laboratory activities involving use of the mathematical programming language ISETL
that students used to write short computer programs. The purpose of the programming
activities is to encourage reflective abstractions, for example, interiorization, by having
students write programs that perform Actions on appropriate input, and encapsulation, by
having students use a program as input and/or output in another program.
Bayazit, I. (2010). The influence of teaching on student learning: The notion of piecewise function.
International Electronic Journal of Mathematics Education, 5(3), 146–164.
This paper examines the influence of classroom teaching on student understanding of
piecewise defined functions. The participants consisted of two experienced mathematics
teachers and their 9th grade students. Using a theoretical framework that emerged from an
APOS analysis, the author illustrated that the teachers differed remarkably in their
approaches to the essence of piecewise functions and that this, in turn, substantially affected
their students’ understanding of this notion. The author found that Action-oriented
200 12 Annotated Bibliography
teaching, which is distinguished by the communication of rules, procedures, and factual
knowledge, confines students’ understanding to an Action conception of piecewise
functions, whereas Process-oriented teaching, which places a priority on conceptual devel-
opment and involves consideration of multiple representations, encourages development of
a Process conception of the function concept.
Bayazit, I., & Gray, E. (2008). Qualitative differences in the teaching and learning of the constant
function. Mediterranean Journal for Research in Mathematics Education, 7, 147–163.
This paper examines two experienced Turkish teachers’ teaching of the constant function
and their students’ resulting understandings. Using a theoretical framework based on APOS
Theory, the authors illustrated that the teachers differed markedly in their approaches to the
essence of the concept. Though their personal subject matter knowledge and understanding
of the potential difficulties and misconceptions associated with the acquisition of aspects of
the function concept were similar, and although they assigned similar tasks, their classroom
presentations focused on qualitatively different aspects of the concept. This had a consid-
erable influence on their students’ construction of knowledge.
Bodı, S., Valls, J., & Llinares, S. (2005). El analisis del desarrollo del esquema de divisibilidad en
N. La construccion de un instrumento. Numeros, 60, 3–24.
The aim of this study is to use APOS Theory to validate an instrument built to evaluate the
development of the comprehension of divisibility. The paper includes an analysis of the
activities and the problems from different textbooks, as well as a review of previous
research on the comprehension of divisibility, that was used to prepare a questionnaire
that includes the mathematical content of the secondary school curriculum. A subsequent
psychometric analysis that was validated by clinical interviews was performed about the
index of difficulty of the questionnaire. The analysis enabled the discrimination of different
ways secondary students understand the notions of divisibility.
Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process
conception of function. Educational Studies in Mathematics, 23, 247–285.
The authors show that APOS Theory, and how it applies to the concept of function, point to
an instructional treatment, using computers, that results in substantial improvements in
students’ understanding of the concept of function. The data analysis shows that students
appear to develop a Process conception of function that they use to perform certain
mathematical tasks.
Brown, A., DeVries, D., Dubinsky, E., & Thomas, K. (1997). Learning binary operations, groups,
and subgroups. Journal of Mathematical Behavior, 16, 187–239.
APOS Theory was used to study students’ learning of binary operations, groups, and
subgroups. The authors propose preliminary genetic decompositions of these topics, describe
an instructional treatment designed to foster development of the proposed mental
constructions, discuss the results of interviews and performance on examinations, suggest
revisions of the genetic decompositions as a result of their analysis of the data, and offer
pedagogical suggestions. The results suggest that the pedagogical approach, based on appli-
cation of the instruction detailed in Dubinsky and Leron (1994), was reasonably effective in
helping students to develop strong conceptions of binary operations, groups, and subgroups.
Brown, A., McDonald, M., & Weller, K. (2010). Step by step: Infinite iterative processes and
actual infinity. In Research in Collegiate mathematics education VII. CBMS issues in mathe-
matics education (Vol. 16, pp. 115–141). Providence, RI: American Mathematical Society.
Students in two introduction to abstract mathematics courses were interviewed while trying
to determine whether the set [1k¼1 P 1; 2; . . . ; kf gð Þequals the set P(N), where N denotes the
set of natural numbers and P denotes the power set operator. An APOS analysis of the data
12.1 A Through B 201
describes the role of interiorization, coordination, and encapsulation in the development of
infinite iterative Processes and their states at infinity. The theoretical analysis is illustrated
through a series of case studies and is compared to what is predicted by the Basic Metaphor
of Infinity of Lakoff and Nunez (2000).
Brown, A., Thomas, K., & Tolias, G. (2002). Conceptions of divisibility: Success and understand-
ing. In S. Campbell & R. Zazkis (Eds.), Learning and teaching number theory: Research incognition and instruction (pp. 41–82). Westport: Ablex Publishing.
The authors report on an examination of prospective elementary teachers’ understanding of
the concept of multiples, with a particular focus on the least common multiple. Students’
understanding is examined using APOS Theory combined with a stage model adapted from
Piaget’s work in Success and Understanding (Piaget 1978).
12.2 C Through De
Carlson, M. (1998). A cross-sectional investigation of the development of the function concept.
Research in Collegiate mathematics education III. CBMS issues in mathematics education (Vol.
7, pp. 114–162). Providence, RI: American Mathematical Society.
In this study, the author investigates students’ development of the function conception. An
exam measuring students’ understandings of major aspects of the function concept was
developed and administered to students who had just received A’s in college algebra,
second-semester honors calculus, or first-year graduate mathematics courses. Follow-up
interviews were conducted with five students from each of these groups. APOS Theory was
one of several theoretical frameworks used to classify students’ conceptual views of
function. The author reaches a number of conclusions, including agreement with
Breidenbach et al. (1992), that students’ understanding of functions was improved as a
result of engaging students in certain types of construction activities.
Cetin, I. (2009). Students’ understanding of limit concept: An APOS perspective. Doctoral Thesis,Middle East Technical University, Turkey.
The main purpose of this study is to investigate first-year calculus students’ understanding
of the formal limit concept and the change in their understanding after following an
instruction designed by the researcher and based on APOS Theory. The case study method
was utilized to explore the research questions. Twenty-five mathematics majors from
Middle East Technical University in Turkey who were taking first-year calculus
participated in the study. The students had five weeks of instruction in the fall semester
of 2007–2008. Each week they met for 2 hours in a computer laboratory to study in groups
and then they attended 4 hours of class. In the computer lab, they worked on programming
activities in order to reflect on the limit concept before they received formal lecture in class.
A questionnaire on limits including open-ended questions was administered as a pretest and
posttest to determine changes in students’ understanding of this concept. At the end of the
instruction, a semi-structured interview protocol developed by the researcher was
administered to all of the students to explore their understanding in depth. The students’
responses in the questionnaire were analyzed both qualitatively and quantitatively.
The interview results were analyzed using the APOS framework. The results of the study
showed that students thinking reflected what was predicted by the preliminary genetic
decomposition. The instruction was found to play a positive role in facilitating students’
understanding of the limit concept.
202 12 Annotated Bibliography
Clark, J. M., Cordero, F., Cottrill, J., Czarnocha, B., DeVries, D. J., St. John, D., Tolias, G., &
Vidakovic, D. (1997). Constructing a schema: The case of the chain rule. Journal of Mathemat-ical Behavior, 16, 345–364.
Based on a preliminary genetic decomposition of how the chain rule concept may be
developed, the authors used APOS Theory together with Piaget and Garcıa’s ideas on
the development of Schema to develop a triad of Schema development that provided a
structure to interpret students’ understanding of the chain rule and to classify their
responses. The results of the data analysis allowed for a revised epistemological analysis
of the chain rule.
Clark, J., Hemenway, C., St. John, D., Tolias, G., & Vakil, R. (1999). Student attitudes toward
abstract algebra. Primus, 9, 76–96.
The authors conducted a comparative study of student attitudes in an abstract algebra
course. One group of students completed an APOS-based instructional sequence that
included computer programming activities and cooperative learning. The other group of
students received traditional instruction. Students from both groups shared their
impressions about the course and of abstract algebra in general in individual interviews
conducted at the conclusion of the course. The students’ responses favored the APOS
approach in many ways, even though the content of the APOS course was at least as
rigorous and demanding as the traditional course.
Cooley, L., Trigueros, M., & Baker, B. (2007). Schema thematization: A theoretical framework
and an example. Journal for Research in Mathematics Education, 38, 370–392.
Although the idea of a thematization of a Schema emerged very early in the development of
APOS Theory (Dubinsky 1986b), it was not studied extensively until this and a related
study (Baker et al. 2000). In these studies, the authors studied the development of a
“Calculus Graphing Schema” via the triad of Schema development. The present investiga-tion builds on this previous work and focuses on the thematization of a Schema. Successful
calculus students were interviewed. They appeared to be operating at different stages of
development of the “Calculus Graphing Schema”. Only one student showed to have
thematized this Schema.
Cordero, F. (1998). El entendimiento de algunas categorıas del conocimiento del calculo y
analisis: el caso de comportamiento tendencial de las funciones. Revista Latinoamericana deInvestigacion en Matematica Educativa, 1, 56–74.
In the school-teaching context, the author encountered an argument given by students on
the subject of graphs of functions. He calls this argument the “tendencial behavior of
functions” because of its nature. The author shows some constructions of this argument that
were made by the student participants and analyzes the data using a version of APOS
Theory.
Cordero, F., & Miranda, E. (2002). El entendimiento de la transformada de Laplace: una
epistemologıa como base de una descomposicion genetica. Revista Latinoamericana deInvestigacion en Matematica Educativa, 5(2), 133–168.
In this paper, two issues are considered: the didactical mathematical discourse related to the
Laplace transform and a theoretical questioning of the notion of genetic decomposition that
could possibly be reformulated with an epistemological basis. The research points to the
absence of a reference frame related to the meaning of this concept and the origin of
the conditions that would allow its construction. This fact questioned any formulation of the
genetic decomposition, since it would imply a learning model for students and a genetic
decomposition formulated in terms of mental constructions to be aware only of the
12.2 C Through De 203
definition of the Laplace transform. Then an epistemology of Laplace transform is
formulated and its role as a basis for a genetic decomposition is discussed with the intention
to enlarge its conceptual frame.
Cottrill, J. F. (1999). Students’ understanding of the concept of chain rule in first year calculus andthe relation to their understanding of composition of functions. Unpublished doctoral disserta-
tion, Purdue University, West Lafayette.
This is a follow-up study to Clark et al. (1997). The author finds that the triad mechanism
describes the observations of student behaviors and can be used to develop instruction to
help students make certain mental constructions. It presents more detailed descriptions of
the Intra-, Inter-, and Trans-levels of the development of the chain rule Schema than those
given in Clark et al. (1997).
Cottrill, J., Dubinsky, E., Nichols, D., Schwingendorf, K., Thomas, K., & Vidakovic, D. (1996).
Understanding the limit concept: Beginning with a coordinated process schema. Journal ofMathematical Behavior, 15, 167–192.
This is the first publication in which the acronym APOS was used as a name for the theory.
The authors reinterpret some points in the literature about the concept of limit and suggest a
new variation of the dichotomy, considered by various authors, between dynamic or
Process conceptions of limits and static or formal conceptions. They also propose
explanations of why these conceptions are so difficult for students to construct. They
present a genetic decomposition for the limit concept and then describe the evolution of
the genetic decomposition using examples from their analysis of interviews with 25 students
from a calculus course.
Czarnocha, B., Dubinsky, E., Prabhu, V., & Vidakovic, D. (1999). One theoretical perspective in
undergraduate mathematics education research. In O. Zaslavsky (Ed.), Proceedings of the 23rdConference of the International Group for the Psychology of Mathematics Education (Vol.
1, pp. 95–110). Haifa, Israel.
The core of this study is the presentation of the belief that research in undergraduate
mathematics education should, on the one hand, be closely connected to curriculum
development and teaching practice and, on the other hand, be used as a source for
empirical data for one or more theories of learning. The study continues to describe
how this belief plays a major role in APOS-based research for both the postsecondary and
K-12 levels. The study ends with a consideration of alternative perspectives to APOS
Theory.
Davis, G., & Tall, D. (2002). What is a scheme? In D. Tall & M. Thomas (Eds.), Intelligence,learning and understanding in mathematics: A tribute to Richard Skemp (pp. 141–160).
Flaxton, QLD: Post Pressed.
This chapter is dedicated to, and fundamentally influenced by, Richard Skemp’s pioneering
work on schemes. The authors discuss examples of scheme formation; schemes and
symbols; schemes as mental Objects; perceptual, social, and conceptual categorization;
and the connection to APOS Theory.
DeVries, D., & Arnon, I. (2004). Solution—What does it mean? Helping linear algebra students
develop the concept while improving research tools. In M. Hoines & A. Fuglestad (Eds.),
Proceedings of the 28th Conference for the International Group for the Psychology ofMathematics Education (Vol. 2, pp. 55–62). Bergen, Norway.
Twelve linear algebra students were interviewed after completing the course about the
concept of a solution of a system of equations. The interviews were analyzed using an
204 12 Annotated Bibliography
APOS genetic decomposition of the topic. The analysis of the interviews revealed several
misconceptions of solution (some of which might be related to misconceptions reported in
the literature on the equality sign). The analysis also revealed shortcomings of the ques-
tionnaire that was used in the interviews: it did not create a distinction between total lack of
knowledge and partial knowledge. Research tools were improved (genetic decomposition,
suggestions for teaching materials, and the questionnaire) and prepared for the next cycle of
research.
12.3 Dubinsky (as Lead Author)
Dubinsky, E. (1984). The cognitive effect of computer experiences on learning abstract mathe-
matical concepts. Korkeakoulujen Atk-Uutiset, 2, 41–47.
This is the first publication concerning Dubinsky’s ideas about incorporating Piaget’s ideas
of reflective abstractions into postsecondary mathematics. In this talk, the author discusses
the distinction between thinking about a function as a Process and as an Object and using
the experience of computer programming to help students understand that distinction. He
also describes the application of Actions to mental Objects, gives an example of a Pascal
program to represent a proof by mathematical induction, and expresses the view that if
students write, debug, and use such a program, their development of an understanding of
induction is enhanced. For the first time, Dubinsky spoke about Actions, Processes, and
Objects; how an external Action is transformed to an internal Process; and how an Action is
applied to a Process or to a mental Object.
Dubinsky, E. (1985, March). Computer experiences as an aid in learning mathematics concepts.Working paper for the Conference on the Influence of Computers and Informatics on Mathe-
matics and its Teaching, Strasbourg.
The term encapsulation, as the mental mechanism for transforming a Process to a mental
Object, is introduced for the first time.
Dubinsky, E. (1986a). On teaching mathematical induction I. Journal of Mathematical Behavior,5, 305–317.
This study presents a prototype version of what was at the time (1986) a novel approach for
teaching mathematical induction. An instructional treatment using computer activities was
introduced in a small class of 8 college students. The instructional treatment, based on an
early version of what would develop into APOS Theory, was designed to help students to
make certain mental constructions through reflective abstraction. Computer activities were
already used to enhance reflective abstractions. The method seemed to be reasonably
effective and several areas of possible improvement were indicated.
Dubinsky, E. (1986b, September 25–27). Reflective abstraction and computer experiences: A new
approach to teaching theoretical mathematics. In Proceedings of the Eighth Annual PME-NAMeeting, East Lansing, MI.
This manuscript introduced interiorization as the mechanism for converting an Action to a
mental Process. The idea of transforming a Schema to an Object, which could be acted on
by another Schema, was also discussed. In this article, the transformation of a Schema to an
Object was treated as an instance of encapsulation, as opposed to the term thematization,
which would arise later.
12.3 Dubinsky (as Lead Author) 205
Dubinsky, E. (1989). On teaching mathematical induction II. Journal of Mathematical Behavior,8, 285–304.
In this paper, a continuation of Dubinsky and Lewin (1986) and Dubinsky (1986a), the
author details two classroom experiments in which a theoretically based instructional
approach (not yet called APOS) using computer experiences with SETL and ISETL was
implemented. The data showed that students seemed to develop a more positive attitude and
were totally successful in solving straightforward induction problems. When presented with
more difficult, unfamiliar problems, they tended to set up most problems correctly, knew
how to use induction, and intended to do so but continued to exhibit difficulty in completing
the proofs.
Dubinsky, E. (1991a). The constructive aspects of reflective abstraction in advanced mathematics.
In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience. New York:
Springer.
The author presents a brief discussion of APOS (not yet named so) as a developing theory
of mathematical knowledge and its acquisition. He also describes specific methods of
construction observed with students. He presents an analysis of studies of induction,
quantification, and function according to this point of view.
Dubinsky, E. (1991b). Reflective abstraction in advanced mathematical thinking. In D. Tall (Ed.),
Advanced mathematical thinking (pp. 95–123). Dordrecht, The Netherlands: Kluwer.
The author makes the case that the concept of reflective abstraction can be a powerful tool
in the study of advanced mathematical thinking, can provide a theoretical basis that
supports and contributes to an understanding of what this thinking is, and suggests how
students can be helped to develop the ability to engage in this type of thinking.
Dubinsky, E. (1992). A learning theory approach to calculus. In Z. Karian (Ed.), Symboliccomputation in undergraduate mathematics education. MAA Notes 24 (pp. 48–55).
Washington, DC: Mathematical Association of America.
The author outlines APOS Theory (not yet named so) and discusses some of the choices
about teaching that seem to follow from the theory. In particular, he discusses how
computers can be used in teaching and learning.
Dubinsky, E. (1994). A theory and practice of learning college mathematics. In A. Schoenfeld
(Ed.), Mathematical thinking and problem solving (pp. 221–243). Hillsdale: Erlbaum.
In mathematics education, there are many dichotomies. The author is interested in binarysyntheses for each of these dichotomies. Specifically he refers to the following dichotomies:
theory and practice, research and development (where development means curriculum
development), and beliefs and choices. In addition to discussing these syntheses, he
describes the beginning of a theory and its application and goes on to describe some
examples of instructional treatments derived from the theory (where the term APOS,
again, has not yet emerged). Results of research that accompanied these treatments suggest
that it might be possible to design instructional treatments based on a theory of learning that
involves use of computers.
Dubinsky, E. (1995). ISETL: A programming language for learning mathematics.
Communications in Pure and Applied Mathematics, 48, 1–25.
The author gives a brief history of the development of a pedagogical strategy for helping
students learn mathematical concepts at the postsecondary level. The method uses ISETL to
implement instruction designed on the basis of APOS Theory (not yet named so). ISETL is
206 12 Annotated Bibliography
described in some detail and examples of its application are given for its use as a pedagogical
tool in abstract algebra, calculus, and mathematical induction.
Dubinsky, E. (1996a). Applying a Piagetian perspective to post-secondary mathematics education.Second International Workshop on Mathematics Education for Engineers (pp. 25–29). Havana.
The author begins with a brief introduction on Piaget’s ideas about education. He shows
how these ideas form the theoretical foundation for curriculum development activities and
inform the overall structure of a particular pedagogical strategy. The author describes how
this constructivist approach uses analysis of data from students to determine the mental
structures that might be needed for the development of mathematical knowledge and gives
examples of how computer activities can be used to help students construct the needed
structures. The manuscript does not report on existing research but refers to relevant
publications.
Dubinsky, E. (1996b). Aplicacion de la perspectiva piagetiana a la educacion matematica
universitaria. Educacion Matematica, 8(3), 24–41.
This report describes Dubinsky’s work on curricular development at the college level. It is
based upon Piaget’s ideas about the way teaching can help a child to learn. The article
begins with a brief introduction to Piaget’s ideas and shows how they form the foundation
of Dubinsky’s activities in curricular development. The author shows how a theoretical
perspective can be used to explain students’ answers to an interview question about the
order of the elements in a group. The author also includes examples of computer tasks that
appear in the activities.
Dubinsky, E. (1997a). On learning quantification. Journal of Computers in Mathematics andScience Teaching, 16(2/3), 335–362.
In this study, the author examines students’ learning of universal and existential
quantification in a specially designed course based on the theoretical analysis of quanti-
fication found in Dubinsky, Elterman, and Gong (1988). The instruction was designed to
assist students in making mental constructions using the computer program ISETL.
Students’ responses to written questions suggest that the pedagogical approach helped
students to develop their understanding of quantification, even when working on
difficult problems.
Dubinsky, E. (1997b). Some thoughts on a first course in linear algebra on the college level. In
D. Carlson, C. Johnson, D. Lay, D. Porter, A. Watkins, & W. Watkins (Eds.), Resources forteaching linear algebra. MAA Notes 42 (pp. 85–106). Washington, DC: Mathematical Asso-
ciation of America.
This chapter is a reaction to the recommendations of two programs for teaching linear
algebra and a proposal for an alternative instructional approach based on APOS Theory.
The proposal includes specific descriptions of how ISETL activities can be incorporated in
the proposed teaching sequence. The study contains detailed descriptions of the three
approaches.
Dubinsky, E. (2000a). Mathematical literacy and abstraction in the 21st century. School Scienceand Mathematics, 100(6), 289–297.
In this paper, the author explains the growing need for abstraction as an important
component of literacy for life in the twenty-first century. He explains how abstraction
occurs and why its development needs to be taught. He reinforces his perspective through
example, specifically of instruction on the mathematical concept of function, and in
12.3 Dubinsky (as Lead Author) 207
consideration of economical/political issues such as the changing rate of change of thenational debt. The author proposes APOS as one possible instructional tool for encouraging
abstraction in postsecondary education and reflects on the necessity to develop similar tools
at the K-12 level.
Dubinsky, E. (2000b). Meaning and formalism in mathematics. International Journal ofComputers for Mathematical Learning, 5, 211–240.
This essay is an exploration of possible psychological sources of mathematical ideas,
specifically the relation between meaning and formalism. Two possible relations between
the two are suggested. Although the use of formalism to construct meaning is very
difficult for students, the author suggests the possibility that this is the only route to
learn large portions of mathematics in upper high school and tertiary levels. The essay
concludes with an outline of APOS as a pedagogical strategy for helping students to deal
with formalism.
Dubinsky, E., & Harel, G. (1992). The nature of the process conception of function. In G. Harel,
E. Dubinsky (Eds.), The concept of function: Aspects of epistemology and pedagogy. MAA
Notes 25 (pp. 85–106). Washington, DC: Mathematical Association of America.
The authors interviewed 13 students who received APOS-based instruction on the concept
of function that included programming activities involving use of ISETL. After examining
the students’ thinking from a number of different perspectives, the authors found that
students have difficulty constructing a Process conception of function because of the
complexity of the construction.
Dubinsky, E., & Leron, U. (1994). Learning abstract algebra with ISETL. New York: Springer.
This is a textbook for a course in Abstract Algebra. It is intended to support APOS Theory
as a constructivist (in the epistemological, not mathematical sense) approach to teaching,
although the term APOS was still not in use at the time. In the introduction (Comments for
the Instructor, pp. xvii–xix), the authors present their constructivist approach: “students
construct, for themselves, mathematical concepts”; “the ideas in the textbook are not
presented in a completed, polished form.” They discuss the use of ISETL and the ACE
Teaching Cycle and consider issues related to the covering of the course material.
Dubinsky, E., & Lewin, P. (1986). Reflective abstraction and mathematics education: The genetic
decomposition of induction and compactness. Journal of Mathematical Behavior, 5, 55–92.
The authors formulate a precursor to APOS Theory by interpreting Piaget’s epistemology,
especially equilibration and reflective abstraction. They apply Piaget’s epistemology to
describe genetic decompositions of mathematical induction and compactness.
Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning in undergrad
mathematics education. In D. Holton (Ed.), The teaching and learning of mathematics at theuniversity level: An ICMI study (pp. 273–280). Dordrecht, The Netherlands: Kluwer.
The authors present six ways in which a theory in mathematics education can contribute to
research and suggest how those ways can be used as criteria for evaluating a theory. Then
they describe how members of RUMEC (Research in Undergraduate Mathematics Educa-
tion Community) and others use APOS Theory to design instruction, develop curricula,
and conduct research as well as explain how the theory is used as a language to communi-
cate ideas about learning. The chapter includes an annotated bibliography that presents
further details about the theory and its use in research in undergraduate mathematics
education.
208 12 Annotated Bibliography
Dubinsky, E., & Schwingendorf, K. (1990). Calculus, concepts, and computers—Innovations in
learning calculus. In T. Tucker (Ed.), Priming the calculus pump: Innovations and resources.MAA Notes 17 (pp. 175–198). Washington, DC: Mathematical Association of America.
This report describes a three-semester innovative calculus course developed at
Purdue University with support from the US National Science Foundation. The materials
were used at Purdue and other universities in the USA and other countries during
the decade of the 1990s. The course is still being used in some universities at the time
of this writing.The design of the course is based on APOS Theory and involves students writing and
running programs in a mathematical programming language and making calculations on the
computer using a symbolic computing system. The pedagogical strategy consists of
students working in cooperative groups in a computer lab where they are expected to
construct mental structures proposed by theoretical analyses of the mathematics they are
trying to learn, small group problem solving in a classroom where students are confronted
with problem situations designed to get them to use the mental structures developed in the
computer lab to construct their understanding of mathematical concepts, and homework
exercises intended to reinforce their understandings and provide practice with standard
calculus problems.The report describes APOS Theory and how it is used in design of the course. Detailed
examples are given of the treatment of graphs and functions, the fundamental theorem of
calculus, infinite sequences, and infinite series. This article includes some of the programs
the students are asked to write, comparative data on students’ learning of calculus, and
comments from students and administrators.
Dubinsky, E., & Schwingendorf, K. (1991a). Constructing calculus concepts: Cooperation in a
computer laboratory. In C. Leinbach, J. R. Hundhausen, A. M. Ostebee, L. J. Senechal, & D. B.
Small (Eds.), The laboratory approach to teaching calculus. MAA Notes 20 (pp. 47–70).
Washington, DC: Mathematical Association of America.
Dubinsky, E., & Schwingendorf, K. (1991b). Calculus, concepts, and computers: Some laboratory
projects for differential calculus. In C. Leinbach, J. R. Hundhausen, A. M. Ostebee, L. J.
Senechal, & D. B. Small (Eds.), The laboratory approach to teaching calculus. MAA Notes
20 (pp. 197–212). Washington, DC: Mathematical Association of America.
This is a pair of papers that extend Dubinsky and Schwingendorf (1990) with a full
discussion of the theoretical background for the laboratory approach and a description of
the cooperative learning environment in the first paper. The second paper provides 14 pages
of sample lab and homework assignments.
Dubinsky, E., Schwingendorf, K. E., & Mathews, D. M. (1995a). Calculus, concepts & computers(2nd ed.). New York: McGraw-Hill.
This is a first-year course in calculus employing the ACE Teaching Cycle using ISETL
activities. This is the textbook used in the course described above in Dubinsky and
Schwingendorf (1990, 1991a, b). The text covers limits, differential and integral calculus,
sequences and series, and polar and parametric curves in a two-semester sequence.
Dubinsky, E., & Schwingendorf, K. E. (1995b). Calculus, concepts, and computers: Multivariableand vector calculus (Revised Preliminary Version). New York: McGraw-Hill.
This is the textbook used in a third semester follow-up to the two-semester course described
above in Dubinsky et al. (1995a). The follow-up course treats the calculus of functions of
several real variables and uses all of the theoretical background and pedagogical strategies
employed in the first two semesters.
12.3 Dubinsky (as Lead Author) 209
Dubinsky, E., & Wilson, R. (2013). High school students’ understanding of the function concept.
Journal of Mathematical Behavior, 32, 83–101.
This paper is a study of part of the Algebra Project’s program for high school students from
the lowest quartile of academic achievement and social and economic status. The study
focuses on learning the concept of function. APOS Theory is used here as a strictly
analytical evaluative tool. Fifteen high school students from the project’s target population
participated in the research. Immediately after instruction, a written instrument was
administered. Several weeks later, in-depth interviews were conducted and then analyzed
using APOS-based tools. The results indicate that with appropriate pedagogy, students from
the project’s target population are able to learn a substantial amount of nontrivial mathe-
matics at the high school level.
Dubinsky, E., & Yiparaki, O. (1996, July 25–26). Predicate calculus and the mathematicalthinking of students, international symposium on teaching logic and reasoning in an illogicalworld (Report). Centre of Discrete Mathematics and Theoretical Computer Science, Rutgers
University.
This report is based on two related projects. The first was an attempt to apply ideas taken
from Piaget to analyze how students might come to understand predicate calculus and how
to apply this analysis in the design and implementation of instruction (Dubinsky et al. 1988;
Dubinsky 1997a). The second project returns to earlier work in an attempt to apply what
today is called APOS Theory (Asiala et al. 1996). This report begins with a statement about
what the authors think about the role of predicate calculus in understanding mathematics
and concludes with the claim that this role does not appear to work for many students. The
authors found that students who participated in an APOS-based implementation did
develop some understanding of quantification and the ability to work with it. This view is
supported by the overall performance of the students in the experiment.
Dubinsky, E., Dautermann, J., Leron, U., & Zazkis R. (1994). On learning fundamental concepts
of group theory. Educational Studies in Mathematics, 27, 267–305.
This is a systematic investigation using APOS Theory of students’ construction of the
concepts of group, subgroup, coset, normality, and quotient group. The authors make
general observations about learning these topics and discuss the complex nature of “under-
standing” and the role of errors and misconceptions.
Dubinsky, E., Elterman, F., & Gong, C. (1988). The student’s construction of quantification. Forthe Learning of Mathematics—An International Journal of Mathematics Education, 8, 44–51.
As part of a course in discrete mathematics, the authors designed a unit on quantification
using computer activities with SETL (the forerunner of ISETL). Using data collected from
students who completed the unit, the authors propose a genetic decomposition for the
concept of quantification.
Dubinsky, E., Weller, K., & Arnon, I. (2013). Preservice teachers’ understanding of the relation
between a fraction or integer and its decimal expansion: The Case of 0.999. . . and 1. CanadianJournal of Science, Mathematics, and Technology Education, 13(3).
This is the third of a sequence of three studies based on data collected from the same
experiment (Weller et al. 2009, 2011). The authors devise a genetic decomposition of
students’ progress in their development of an understanding of the decimal 0:9 and its
relation to 1. The genetic decomposition appears to be valid for a high percentage of the
study participants and suggests the possibility of a new stage in APOS that would be
the first substantial change in the theory since its inception (Dubinsky and Lewin 1986).
The analysis includes a relatively objective and highly efficient methodology that might be
210 12 Annotated Bibliography
useful in other research and in assessment of student learning. For further analysis of this
study, see Weller et al. (2009, 2011) in this bibliography.
Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005a). Some historical issues and
paradoxes regarding the concept of infinity: An APOS analysis: Part 1. Educational Studies inMathematics, 58, 335–359.
This paper applies APOS Theory to suggest a new explanation of how people might think
about the concept of infinity. The authors propose cognitive explanations and, in some
cases, resolutions of various dichotomies, paradoxes, and mathematical problems involving
the concept of infinity. These explanations are expressed in terms of the mental
mechanisms of interiorization and encapsulation. The purpose for providing a cognitive
perspective is that issues involving the infinite have been and continue to be a source of
interest, of controversy, and of student difficulty.
Dubinsky, E., Weller, K., McDonald, M. A., & Brown, A. (2005b). Some historical issues and
paradoxes regarding the concept of infinity: An APOS analysis: Part 2. Educational Studies inMathematics, 60, 253–266.
This is the second part of a study on how APOS Theory may be used to provide cognitive
explanations of how students and mathematicians might think about the concept of infinity.
The authors discuss infinite Processes, describe how the mental mechanisms of interioriza-
tion and encapsulation can be used to conceive of an infinite Process as a completed totality,
explain the relationship between infinite Processes and the Objects that may result from
them, and apply their analyses to certain mathematical issues related to infinity.
Dubinsky, E., Weller, K., Stenger, C., & Vidakovic, D. (2008). Infinite iterative processes: The
tennis ball problem. European Journal of Pure and Applied Mathematics, 1(1), 99–121.
In this paper, the authors use APOS Theory to describe the mental constructions needed to
understand and to solve the Tennis Ball Problem, a paradoxical problem that involves the
coordination of three infinite Processes. Of the 15 interview subjects, only one solved the
problem correctly. His responses indicated that he had made the mental constructions called
for by the theoretical analysis, while the other students gave evidence of not having made
those constructions. On the basis of the data analysis, the authors offer various pedagogical
suggestions and avenues for future research.
12.4 E Through M
Ely, R. (2011). Envisioning the infinite by projecting finite properties. Journal of MathematicalBehavior, 30, 1–18.
Twenty-four postsecondary students were interviewed as they worked on the Tennis Ball
Problem. The author presents a framework for making sense of the participants’ responses.
The author does not assume that this framework is a universal or comprehensive framework
to describe the learning of infinite Processes, but suggests that other existing frameworks,
such as the BMI1 and APOS, are limited in accounting for the participants’ responses.
1 For BMI, see: Lakoff, G., & Nunez, R. (2000).Where mathematics comes from. New York: Basic
Books.
12.4 E Through M 211
Fenton, W., & Dubinsky, E. (1996). Introduction to discrete mathematics with ISETL. New York:
Springer.
Intended for first- or second-year undergraduates, this introduction to discrete mathematics
covers the usual topics of such a course but applies constructivist principles that promote—
indeed, require—active participation by the student. Working with the programming
language ISETL, whose syntax is close to that of standard mathematical language, the
student constructs the concepts in her or his mind as a result of constructing them on the
computer in the syntax of ISETL. This dramatically different approach allows students to
attempt to discover concepts in a “Socratic” dialog with the computer. The discussion
avoids the formal “definition-theorem” approach and promotes active involvement by the
reader by its questioning style. An instructor using this text can expect a lively class whose
students develop a deep conceptual understanding rather than simply manipulative skills.
Font, V., Malaspina, U., Gimenez, J., &Wilhelmi, M. R. (2011). Mathematical objects through the
lens of three different theoretical perspectives. In E. Svoboda (Ed.), Proceedings of the SeventhCongress of the European Society for Research in Mathematics Education. Rzeszow.
In this paper, a link between the onto-semiotic approach (OSA) to mathematics cognition
and instruction, APOS Theory, and the cognitive science of mathematics (CSM) is
established as it regards use of the concept “mathematical object.” It is argued that the
notion of object used in the OSA does not contradict that employed by APOS Theory or the
CSM, since the latter two theories highlight partial aspects of the complex process through
which, according to the OSA, mathematical objects emerge out of mathematical practices.
Gavilan, J. M., Garcıa, M. M., & Llinares, S. (2007a). La modelacion de la descomposicion
genetica de una nocion matematica. Explicando la practica del profesor desde el punto de vista
del aprendizaje potencial en los estudiantes. Educacion Matematica, 19(2), 5–39.
The construct “modeling of the genetic decomposition of a notion” is introduced in order to
explain mathematics teachers’ practices from the point of view of the construction of
mathematical knowledge that seems to develop in students. This construct is used to
analyze two teachers’ teaching when introducing the notion of derivative to high school
students (16–18 years). The results of the analysis allowed characterization of the principles
on which the teacher drew in her or his practice. Reflections about this construct are made.
Gavilan, J. M., Garcıa, M. M., & Llinares, S. (2007b). Una perspectiva para el analisis de la
practica del profesor de matematicas. Implicaciones metodologicas. Ensenanza de las Ciencias,25(2), 157–170.
In this study, the authors emphasize that analysis of mathematics teachers’ practice
involves making explicit a model of students’ learning (construction of mathematical
knowledge) and generating analytic tools that allow the explanation of teachers’ practice
in a way coherent with the chosen learning model. In this article, the following notions are
introduced: the analytic tool “modeling of mechanisms for the construction of knowledge”
in order to carry out this analysis and the “vignette” notion as a way to make it explicit in
the analysis of teachers’ practice. The description and interpretation of this practice is based
on two elements: the notion of practice, which provides the sociocultural focus, and APOS
Theory, which offers a theoretical model of knowledge construction.
Hahkioniemi, M. (2005). Is there a limit in the derivative?—Exploring students’ understanding of
the limit of the difference quotient. Proceedings of CERME 4, 1758–1767.
Task-based interviews with five postsecondary students were arranged to investigate
students’ understanding of the limit of the difference quotient (LDQ). The students’
procedural knowledge was analyzed using APOS Theory, and their conceptual knowledge
212 12 Annotated Bibliography
was analyzed by examining the kind of representations they had of the limiting process and
how these were connected to LDQ. It was found that students had two kinds of connections:
change from one representation to other or the explanation of one representation with the
other. Among the students, all combinations of good or poor procedural and conceptual
knowledge of LDQ were found.
Hamdan, M. (2006). Equivalent structures on sets: Equivalence classes, partitions and fiber
structures of functions. Educational Studies in Mathematics, 62, 127–147.
This study reports on how students can be led to make meaningful connections between
structures on a set such as a partition, the set of equivalence classes determined by an
equivalence relation, and the fiber structure of a function on that set (i.e., the set of
pre-images of all sets {b} for b in the range of the function). The author presents an initial
genetic decomposition, in the sense of APOS Theory, for these concepts, and suggests and
applies instructional procedures that reflect the proposed genetic decomposition. The
author suggests the need for a revised genetic decomposition based on informal interviews
with students at different stages in their learning of these concepts.
Harel, G., & Dubinsky, E. (1991). The development of the concept of function by preservice
secondary teachers: From action conception to process conception. In F. Furinghetti (Ed.),
Proceedings of the Fifteenth Conference of the International Group for the Psychology ofMathematics Education (Vol.2, pp. 133–140). Assisi, Italy.
A group of 22 students participated in a course in discrete mathematics using an instruc-
tional treatment based on the constructivist theory that was later named APOS. This
instructional treatment was meant to help the students improve their conceptions of
function. Their starting points ranged from very primitive conceptions to Action
conceptions. As a result of the instructional treatment, all of the students progressed toward
a Process conception of function. The authors list four factors that play a role in the
progression to a full Process conception.
Hernandez Rebollar, L. A., & Trigueros, M. (2012). Acerca de la comprension del concepto de
supremo. Revista educacion Matematica, 24(3).
The main goal of this work was to study how university students construct the supremum
concept. The authors used APOS Theory as a theoretical framework and presented a genetic
decomposition of the supremum concept. To validate the genetic decomposition, the
authors designed a questionnaire for mathematics and physics students at a public univer-
sity and analyzed the data using the theory. Results revealed that most of the students
involved in the study did not construct an Action conception of this concept. The analysis
has been useful in explaining the difficulties students face when they try to demonstrate that
a number is the supremum of a given set.
Hollebrands, K. F. (2003). High school students’ understandings of geometric transformations in
the context of a technological environment. Journal of Mathematical Behavior, 22, 55–72.
This study investigated the nature of students’ understandings of geometric
transformations—translations, reflections, rotations, and dilations. Instruction involved
use of The Geometer’s Sketchpad. The author implemented a seven-week instructional
unit on geometric transformations within an honors geometry class. Students’ conceptions
of transformations as functions were analyzed using APOS Theory. The analysis suggests
that students’ understandings of key concepts such as domain, variables, and parameters, as
well as relationships and properties of transformations, were crucial in the support of deeper
understandings of transformations as functions.
12.4 E Through M 213
Kabael, T. (2011). Generalizing single variable functions to two-variable functions, function
machine and APOS. Educational Sciences: Theory and Practice, 11(1), 484–499.
The study examines how students generalize the concept of function from the single-
variable case to the two-variable case. The author uses APOS to analyze data collected
from 23 students in an Analysis II course in an elementary mathematics education program.
As a result of the data analysis, the author concludes that construction of the two-variable
function concept depends on understanding the one-variable function concept and devel-
opment of a Schema for three-dimensional space.
Ku, D., Oktac, A., & Trigueros, M. (2011). Spanning set and span—An analysis of the mental
constructions of undergraduate students. In S. Brown, S. Larsen, K. Marrongelle, &
M. Oehrtman (Eds.), Proceedings of the 14th annual conference on research in undergraduatemathematics education (pp. 176–186). Washington, DC: Special Interest Group of the Mathe-
matical Association of America (SIGMAA) for Research in Undergraduate Mathematics
Education.
The authors present a genetic decomposition for the construction of the concepts of
spanning set and span in Linear Algebra. They used the genetic decomposition to analyze
data from interviews with 11 students who completed an introductory linear algebra course.
The authors concluded that it is easier in general for students to decide whether a given set
spans a given vector space than to construct a spanning set for a given vector space. Some
modifications to the preliminary genetic decomposition are suggested.
Ku, D., Trigueros, M., & Oktac, A. (2008). Comprension del concepto de base de un espacio
vectorial desde el punto de vista de la teorıa APOE. Educacion Matematica, 20 (2), 65–89.
The authors use APOS Theory to develop a genetic decomposition of the concept of basis.
They test the genetic decomposition empirically by interviewing six undergraduate
students who completed a linear algebra course. The results showed that it was easier for
the students to determine if a given set is a basis of a vector space than to find a basis for a
given vector space. The authors attribute the difference to students’ inability to coordinate
the Processes for linear independence and spanning set.
Llinares, S., Boigues, F., & Estruch, V. (2010). Desarrollo de un esquema de la integral definida en
estudiantes de ingenierıas relacionadas con las ciencias de la naturaleza. Un analisis a traves de
la logica Fuzzi. Revista Latinoamericana de Investigacion en Matematica Educativa, 13,255–282.
This research describes the triad development of a Schema for the concept of definite
integral. Data for the study was gathered from earth science engineering students who were
using fuzzy metrics. The results demonstrate students’ difficulty in linking a succession of
Riemann sums to the limit, which forms the basis for the meaning of the definite integral.
Mathews, D., & Clark, J. (1997, March). Successful students’ conceptions of mean, standarddeviation, and the Central Limit Theorem. Paper presented at the Midwest Conference on
Teaching Statistics, Oshkosh, WI.
The authors present an APOS-based analysis of audiotaped clinical interviews with college
freshmen immediately after they completed an elementary statistics course and obtained a
grade of “A.” The authors found that APOS is a useful way of describing students’
understanding of mean, standard deviation, and the Central Limit Theorem. In addition,
they conclude that traditional instruction in statistics does not help students make the
mental constructions appropriate for development of these concepts. In particular, tradi-
tional instruction seems to inhibit students from moving from a Process to an Object
conception of standard deviation.
214 12 Annotated Bibliography
Mamolo, A. (2009). Accommodating infinity: A leap of imagination. In Proceedings of the 31stannual meeting of the North American Chapter of the International Group for the Psychologyof Mathematics Education (Vol. 5, pp. 65–72). Atlanta, GA: Georgia State University.
This paper presents first results of a study which seeks to identify the necessary and
sufficient features of accommodating the idea of actual infinity. Data was collected from
university mathematics majors’ and graduates’ engagement with the Ping-Pong Ball
Conundrum. APOS Theory was used in the analysis of the data. The paper focuses on the
following feature: the leap of imagination required to conceive of actual infinity and its
associated challenges.
Mamolo, A., & Zazkis, R. (2008). Paradoxes as a window to infinity. Research in MathematicsEducation, 10(2), 167–182.
This study examines approaches to infinity of two groups of university students with
different mathematical background: undergraduate students in Liberal Arts Programs and
graduate students in a Mathematics Education Master’s Program. Data are drawn from
students’ engagement with Hilbert’s Grand Hotel paradox and the Ping-Pong Ball Conun-
drum. Two frameworks were used for the interpretation of students’ responses as well as
their emergent ideas of infinity: reducing abstraction (Hazzan 1999) and APOS. While
graduate students found the resolution of Hilbert’s Grand Hotel paradox unproblematic,
responses of students in both groups to the Ping-Pong Ball Conundrum were surprisingly
similar. Consistent with prior research, the work of participants in this study revealed that
they perceive infinity as an ongoing Process, rather than a completed one, and fail to notice
conflicting ideas. The contribution of this work is in describing specific challenging
features of these paradoxes that might influence students’ understanding of infinity, as
well as the persuasive factors in students’ reasoning, that have not been unveiled by
other means.
Martınez-Planell, R., & Trigueros, M. (2012). Students’ understanding of the general notion of a
function of two variables. Educational Studies in Mathematics, 81, 365–384.
In this study, the authors continue their research on the different components of students’
understanding of two-variable functions. In particular, they consider students’ understand-
ing of the concepts of domain, the possibility of an arbitrary nature of function, the
uniqueness of function image, and range. (Trigueros and Martınez-Planell, 2010) The
thinking of 13 college students was analyzed using APOS Theory and a semiotic represen-
tation theory. The authors concluded that many of the students’ notions of function could be
considered pre-Bourbaki.
McDonald, M., Mathews, D., & Strobel, K. (2000). Understanding sequences: A tale of two
objects. In Research in Collegiate mathematics education IV. CBMS issues in mathematics
education (Vol. 8, pp. 77–102). Providence, RI: American Mathematical Society.
The authors used APOS Theory to examine how students construct the concept of
sequence. The authors show that students tend to construct two distinct cognitive Objects
and refer to both as a sequence. One construction, which the authors call SEQLIST, is what
one might understand as a listing representation of a sequence. The other, which they call
SEQFUNC, is what one might interpret as a functional representation of a sequence. In this
paper, the authors detail students’ constructions of SEQLIST and SEQFUNC and charac-
terize the connections between them using the detailed descriptions of the Intra-, Inter-, and
Trans-levels of the development of the chain rule Schema triad introduced by Clark
et al. (1997).
12.4 E Through M 215
Meel, D. (2003). Models and theories of mathematical understanding: Comparing Pirie and
Kieren’s model of the growth of mathematical understanding and APOS Theory. In Researchin Collegiate mathematics education V. CBMS issues in mathematics education (Vol. 12, pp.
132–187). Providence, RI: American Mathematical Society.
This paper focuses on two theoretical frameworks for understanding student thinking in
mathematics: Pirie and Kieren’s model of the growth of mathematical understanding
(known as The Onion Model) and Dubinsky’s APOS Theory. The author explains how
these two perspectives satisfy criteria for classification as a theory and discusses a variety of
interconnections between these two theories.
Mena, A. (2011). Estudio epistemologico del teorema del isomorfismo de grupos. Doctorate thesis,Cicata-IPN, Mexico.
The author presents a pedagogical approach for the teaching of the isomorphism theorem
for groups that is based on a version for sets followed by a version that incorporates the
group structure. The author proposes a genetic decomposition for construction of the
theorem that involves development of a Schema structure.
Moreira, R. N., & Wodewotzki, M. L. L. (2004). A perspective on the conceptions of college
freshmen regarding absolute value of real numbers. Boletim de Educacao Matematica, 17(22),63–81.
The authors start by discussing how students try to understand the concept of absolute
value. Based on an initial cognitive model, the authors attempt to interpret interview data
using APOS Theory. The results of the analysis seem to suggest that starting college
students’ ability to make abstractions enables them to develop an adequate understanding
of the absolute value concept. The analysis also pointed out that graphical
representations and cooperative learning were relevant factors in the students’ learning
of the concept.
12.5 Works of Piaget
Although these items do not mention APOS Theory, much of the foundation for the
creation and development of APOSTheory comes fromworks of Piaget such as these.
Beth, E. W., & Piaget, J. (1974). Mathematical epistemology and psychology (W. Mays, Trans.).
Dordrecht, The Netherlands: D. Reidel. (Original work published 1966).
The book is in two parts. The first part was written by Beth and the second by Piaget. The
second part is a major source for the foundation of APOS Theory. Piaget argues for a
constructivist epistemology, which he calls genetic epistemology, and explains why it is
superior to other theories such as Platonism, empiricism, apriorism, logical reductionism,
and nominalism (linguistic). He explains how the development of an individual’s mathe-
matical thought applies to advanced mathematical thinking as well as to the thinking of
children. In particular, he relates some of his ideas to the three “mother structures” of
Bourbaki. The key ingredient of genetic epistemology is reflective abstraction, which is
discussed throughout the book and involves actions, operations (processes), and objects.
216 12 Annotated Bibliography
Piaget, J. (1975). Piaget’s theory (G. Cellerier & J. Langer, Trans.). In P.B. Neubauer (Ed.), Theprocess of child development (pp. 164–212). New York: Jason Aronson.
Here Piaget describes his genetic epistemology which was developed over more than half a
century and continued until his death in 1980. The survey begins with a discussion of the
biological origins of the cognitive functions involving several biological and cognitive
mechanisms such as adaptation, assimilation, accommodation, and equilibration. These
bio-cognitive mechanisms are the sources of the cognitive structures that, according to
Piaget, produce all knowledge and intelligence. The paper reiterates Piaget’s belief in the
continuity of the development of thought from infancy to adult scientific thinking. Piaget
describes in some detail his theory of stages, which relates to thinking at the early ages and
then goes on to consider the logico-mathematical aspects of mental structures and their
construction at all ages.
Piaget, J. (1976). The grasp of consciousness (S. Wedgwood, Trans.). Cambridge, MA: Harvard
University Press. (Original work published 1974).
Piaget calls this book the completion of his long study of causality. The main issue he
discusses here is the relation between a child’s ability to complete a task and her or his
understanding of how that task is completed. It turns out that there is a considerable time
delay, observable in many experiments reported in the book, between the former and the
latter. Piaget’s explanation is that the subject takes time to make the mental constructions
by which he or she develops an understanding of the success in performing a task. He
analyzes the mental constructions his subjects appear to be making as he gradually moves
from students who succeeded in the action without consciousness of it to students with
growing levels of cognizance.The format of the book is a description and analysis of 15 different experiments
followed by a conclusion.
Piaget, J. (1978). Success and understanding (A. J. Pomerans, Trans.). Cambridge, MA: Harvard
University Press. (Original work published 1974).
Much of Piaget’s contributions to learning consist of theoretical descriptions of how
knowledge and intelligence develop in the mind of an individual. But his work is far
from purely theoretical. Piaget also wrote several books that report on his empirical studies.
Success and Understanding is one of these. It consists of thirteen chapters, the first 12 of
which report on 12 different experiments and a final chapter in which he summarizes the
first 12 and presents his general conclusions. Each of the first 12 chapters displays Piaget’s
remarkable ability to construct interesting tasks that use “apparatuses” cleverly crafted out
of locally purchased materials that embodied the particular concepts he wished to study.
The subjects, who were children of different ages, engaged in the activities, and Piaget
reported on the successes they did or did not have. Then he interviewed each subject to
understand the subject’s thinking. An interesting theme that runs through the entire book is
that young children very often succeed with a task long before they understand why they
succeeded. Piaget’s analyses of the interview transcripts, many of which appear in the book,
give rise to the points in his theoretical descriptions. One can learn several different things
from reading this book in addition to getting a better understanding of Piaget’s epistemol-
ogy and its source in empirical data. For example, the tasks themselves are powerful tools
for analyzing the thinking of children, and the interview excerpts teach us much about how
to conduct in-depth interviews with children at various ages. Finally, when his often subtle
and opaque theoretical points are couched in the concrete activities of children, they
sometimes become a little easier to understand.
12.5 Works of Piaget 217
Piaget, J., & Garcıa, R. (1989). Psychogenesis and the history of science (H. Feider, Trans.). NewYork: Columbia University Press. (Original work published 1983).
The main purpose of this book is to compare the historical development of scientific
thought to its cognitive development in the mind of an individual. The concern of the
authors is not with the content of concepts but the common mechanisms by which they are
constructed mentally. According to their analysis, both scientific and cognitive
developments proceed by the mental construction of a sequence of stages. In some cases,
the developmental sequences in history and in cognition are parallel, but in other cases, they
are different, even in at least one case, directly opposite. On the other hand, the authors
argue, and support their arguments by reference to empirical evidence, that the basic
mechanisms for mental constructions in history and cognition are the same.These mechanisms are reflective abstraction; an interaction between subject and object
in which experience arises out of interpretation and construction; differentiation and
integration; a search for “reasons,” which means relating phenomena to a mental structure
or coordinated schema; and a sequence of stages each of which is made possible by the
preceding ones and each of which in turn prepares those that follow. The transitional
mechanisms for this sequence of stages exhibit two characteristics that are common
between the history of science and psychological development.The first common transitional mechanism is that each stage is integrated in the
succeeding structure. The second is a new mechanism introduced for the first time in this
book. It is a dialectical triad that leads from intra- (object analysis) to inter- (analyzing
relations or transformations) to trans- (building of structures) levels of analysis.Piaget and Garcia apply these very general considerations to several topics: the devel-
opment of mechanics in physics from Aristotelian to Newtonian thinking, geometry, and
algebra.
12.6 P Through T
Parraguez, M., & Oktac, A. (2010). Construction of the vector space concept from the viewpoint of
APOS theory. Linear Algebra and its Applications, 432, 2112–2124.
APOS Theory is used to propose a genetic decomposition of the vector space concept.
Empirical results are based on an analysis of interview and questionnaire data with
10 undergraduate mathematics students. The analysis focuses on the coordination between
the two operations that form the vector space structure and the relation of the vector space
Schema to other concepts such as linear independence and binary operations.
Pegg, J., & Tall, D. (2005). The fundamental cycle of concept construction underlying various
theoretical frameworks. International Reviews on Mathematical Education (Zentralblatt furDidaktik der Mathematik), 37, 468–475.
In this paper, the authors consider the development of mathematical concepts over time.
Specific attention is given to the shifting of the learner’s attention from step-by-step
procedures that are performed in time to symbolism that can be manipulated as mental
entities on paper and in the mind. The analysis uses different theoretical perspectives,
including the SOLO model, APOS Theory, and various other theories of concept
construction. The analysis reveals a fundamental cycle underlying conceptual develop-
ment from Actions in time to concepts that can be manipulated as mental entities. This
cycle appears widely in different ways of thinking that occur throughout mathematical
learning.
218 12 Annotated Bibliography
Possani, E., Trigueros, M., Preciado, J. G., & Lozano, D. (2010). Use of models in the teaching of
linear algebra. Linear Algebra and its Applications, 432, 2125–2140.
The authors present the results of an approach to teaching linear algebra using models.
Their interest lies in analyzing the use of two theories of mathematics education, namely,
Models and Modeling and APOS Theory. These two theories are used in the design of a
teaching sequence that starts with presenting “real-life” decision-making problems to
students. The possibilities of this methodology are illustrated through the analysis and
description of classroom experience involving a problem related to traffic flow that
elicits the use of a system of linear equations and different parameterizations of this
system to answer questions on traffic control. Cycles of students’ work on the problem
and the advantages of this approach in terms of students’ learning are described. The
possibilities for extending it to other problems and linear algebra concepts are
also discussed.
Ramirez, A. (2009). A cognitive approach to solving systems of linear equations. Ph.D. Disserta-tion, Illinois State University.
In this study, the author investigated the ways in which students come to understand
systems of linear equations. Data were collected from observations of a teaching sequence
with a small linear algebra class and from written tasks presented during interviews with
four students from the same class. In her analysis of the data, the author used APOS Theory
to conclude that systems of linear equations are a part of one’s Object conception of
equivalent systems. This finding represented a modification of the author’s preliminary
genetic decomposition.
Reynolds, B. E., & Fenton, W. E. (2006). College geometry: Using the geometer’s sketchpad.Hoboken, NJ: Wiley.
In this book, APOS Theory is used as a grounded learning theory for college-level courses
on Euclidean and non-Euclidean geometries. The textbook relies on the use of Geometer’s
Sketchpad that provides a dynamic interactive environment for students to explore the
properties of geometric figures and their relationships.
Reynolds, B. E., Przybylski, J., Kiaie, C. C., Schwingendorf, K. E., & Dubinsky, E. (1996).
Precalculus, concepts & computers. New York: McGraw-Hill.
This is a course in precalculus employing the ACE Teaching Cycle using ISETL activities.
This is part of the calculus series with Dubinsky et al. (1995) and Dubinsky and
Schwingendorf (1995).
Roa Fuentes, S. (2012). El infinito: un analisis cognitivo de ninos y jovenes talento en matematicas. Doctorate thesis, Cinvestav-IPN, Mexico.
In this research study, an approach to mathematical talent is presented from a cognitive
point of view, based on three pillars: APOS Theory, the construction of mathematical
infinity as an iterative Process, and academic programs in Colombia and Mexico that focus
on maximizing mathematical talent.A genetic decomposition of infinity is presented, where the mental structures and
mechanisms that an individual might develop in order to construct mathematical infinity
in different contexts are described. Particular analyses for the Tennis Ball Paradox, the
Hilbert’s Hotel, and the construction of the Koch curve are proposed. In all of these
situations, the specificity of the iterative Processes and the role of the context in the
construction of the infinity concept are analyzed.
12.6 P Through T 219
Roa-Fuentes, S., & Oktac, A. (2010). Construccion de una descomposicion genetica: Analisis
teorico del concepto transformacion lineal. Revista Latinoamericana de Investigacion enMatematica Educativa, 13(1), 89–112.
In this article, two preliminary genetic decompositions for the linear transformation
concept are presented in detail, one that aligns with the treatment of this concept in most
textbooks and another that starts with the construction of the concept of (general) transfor-
mation between vector spaces.
Roa-Fuentes, S., & Oktac, A. (2012). Validacion de una descomposicion genetica de
transformacion lineal: Un analisis refinado por la aplicacion del ciclo de investigacion de
APOE. Revista Latinoamericana de Investigacion en Matematica Educativa, 15(2), 199–232.
In this paper, the third component of APOS Theory, namely, the “collection and analysis of
data” phase is developed for the linear transformation concept. The authors design a
diagnostic test and an interview with college students that are based on the theoretical
analysis suggested by Roa-Fuentes and Oktac (2010). Analysis of data shows that the
properties of addition of vectors and multiplication of a vector by a scalar must be
coordinated in order for the student to construct the concept of linear transformation as a
Process. A refined genetic decomposition and didactic suggestions in relation to the
construction of properties and the preservation of linear combinations follow.
Salgado, H., & Trigueros, M. (2009). Conteo: una propuesta didactica y su analisis. EducacionMatematica, 21, 91–117
This paper uses APOS Theory to study the learning of the concepts related to combinations
and permutations. The authors present a preliminary genetic decomposition for the con-
struction of these concepts and consider a didactical approach to teach them at the
university level. After teaching a course, the authors refined the genetic decomposition
and the didactical sequence and tested them in the next semester. The authors analyzed the
production of the students during the two semesters and the results of the exam
corresponding to that topic. Results show how students’ mental constructions develop
while they work with the activities and conclude that students of the second experience
showed a better understanding of the concepts related to combinations and permutations.
Schwingendorf, K. E., McCabe, G. P., & Kuhn, J. (2000). A longitudinal study of the C4L calculus
reform program: Comparisons of C4L and traditional students. In Research in Collegiatemathematics education IV. CBMS issues in mathematics education (Vol. 8, pp. 63–76). Provi-
dence, RI: American Mathematical Society.
The authors present results of a statistical comparison between 205 students who took the
course Calculus, Concepts, Computers, and Cooperative Learning (a reform course designed
using APOS Theory) and 4431 students who took a traditional calculus course at Purdue
University. When compared with the traditionally taught students, the students who received
the reform course earned higher grades in further calculus courses, were as well prepared for
math courses beyond calculus, as well as all other academic courses, took more calculus
courses, and completed about the same number of non-calculus mathematics courses.
Stenger, C., Weller, K., Arnon, I., Dubinsky, E., & Vidakovic, D. (2008). A search for a
constructivist approach for understanding the uncountable set P(N). Revisto Latinoamericanode Investigacion en Matematicas Educativas, 11(1), 93–126.
This study considers the question of whether individuals build mental structures for the set
P(N ) that give meaning to the phrase “all subsets of N.” The contributions concerning this
question are twofold. First, constructivist perspectives were identified and described, such
that have been or could be used to describe individuals’ thinking about infinite sets,
220 12 Annotated Bibliography
specifically the set of natural numbers N. APOS was one of the perspectives considered.
Second, to determine whether individuals’ thinking about the set P(N ) can be interpreted in
terms of one or more of these perspectives, eight mathematicians were interviewed. Their
ideas about N and P(N ) were analyzed in terms of the chosen perspectives. The authors
found that APOS Theory seems to explain more readily the mental structures for N. ForP(N ), the same analysis cast doubt on whether individual understanding of the set P(N )
extends beyond the formal definition.
Tabaghi, S. G., Mamolo, A., & Sinclair, N. (2009). The effect of DGS on students’ conception of
slope. In Proceedings of the 31st annual meeting of the North American Chapter of theInternational Group for the Psychology of Mathematics Education (Vol. 5, pp. 226–234).
Atlanta, GA: Georgia State University.
This report is the first installment of a broader study that investigated university students’
conceptualizations of static and dynamic geometric entities. In this part, a refined look at
the conceptualizations of two groups of students is offered—one group which was taught
using Dynamic Geometric Software and the other in a “traditional” fashion. Both APOS
Theory and the notion of reification were used to interpret learners’ understanding of the
slope of lines. Data revealed that students using DGS developed a strong proceptual
understanding of slope, which enabled them to solve problems in which slope could be
seen as a conceptual Object. This report sets the stage for a look forward to how DGS may
influence learners’ Process-Object conceptualization of other geometric representations of
algebraic equations.
Thomas, K. S. C. (1995). The fundamental theorem of calculus: An investigation into students’constructions. Unpublished doctoral dissertation, Purdue University, West Lafayette.
This study was designed to investigate the question, “How can the fundamental theorem of
calculus be learned, and how do computer activities and the pedagogy of a particular kind
of nontraditional calculus course affect this learning?” The nontraditional calculus course
was based on APOS Theory and used the ACE Teaching Cycle. The study found that the
participants’ function Schemas contained a misconception. The students believed that the
name of the independent variable was a significant characteristic of a function and that it
was a characteristic which was subject to being changed when a Process such as differenti-
ation was applied to the function.
Tossavainen, T. (2009). Who can solve 2x¼1?—An analysis of cognitive load related to learning
linear equation solving. The Montana Mathematics Enthusiast, 6(3), 435–448.
Using 2x ¼ 1 as an example, the cognitive load related to learning how to solve linear
equations is discussed. Intrinsic cognitive loads needed in arithmetical, geometrical, and
real analytical approaches to linear equation solving are considered using the framework of
the Cognitive Load Theory. This is done from the point of view of the conceptual and
procedural knowledge of mathematics and APOS Theory. A design of a setting for teaching
linear equation solving is offered.
Trigueros, M. (2004). Understanding the meaning and representation of straight line solutions of
systems of differential equations. In D.E. McDougall & J.A. Ross (Eds.), Proceedings of theTwenty-sixth Annual meeting of the North American Chapter of the International Group for thePsychology of Mathematics Education (Vol. 1, pp. 127–134). Toronto.
The main purpose of this study is the analysis of student responses to questions related to
their understanding of the meaning and representation of straight-line solutions of systems
of differential equations. Students’ responses to questions involving the linearity theorem in
the context of systems of linear differential equations and the geometric representation of
straight line solutions to these systems were analyzed using APOS Theory with particular
12.6 P Through T 221
focus on the development of Schema structures. Students’ responses provided evidence of
difficulties in relating concepts that come from different areas of mathematics even when
the students could apply certain solution methods. Some instructional activities that seem to
be successful are suggested.
Trigueros, M. (2005). La nocion del esquema en la investigacion en matematica educativa a nivel
superior. Educacion Matematica, 17 (1), 5–31.
Piaget’s work is the epistemological source of some of the theories that are used in the field
of mathematics education research. In this paper, the fundamental ideas of one of these
theories, APOS Theory, are presented. It is shown how this theory is evolving dynamically
and continuously through the investigation of university students’ understanding of
advanced mathematical concepts and whether students are able to integrate several
concepts in the solution of specific problem situations.
Trigueros, M., & Campero, J. (2010). Propuesta didactica en optimizacion dinamica. Investigacion
en el aula. Educacion Matematica, 22(3), 87–117.
The purpose of this paper is to present the results of a research study on a didactical
proposal to teach dynamical optimization, in particular, calculus of variations. The proposal
design was based on APOS Theory and was tested at a private Mexican university. Results
obtained from the analysis of students’ responses to a questionnaire and an interview show
that students construct Process conceptions, and in some cases, Object conceptions, of the
related concepts. However, some obstacles were difficult for the students to overcome.
Trigueros Gaisman, M., & Escandon, C. (2008). Los conceptos relevantes en el aprendizaje de la
graficacion. Un analisis a traves de la estadıstica implicativa. Revista Mexicana de InvestigacionEducativa, 13, 59–85.
Various studies show that students experience difficulties in understanding specific
concepts of differential calculus. Some studies point to the obstacles students have in
integrating different concepts into solving specific problems, including the writing of
functions. The current study uses an instrument based on a genetic decomposition that
was used in previous APOS studies (Cooley et al. 2007). Responses from 40 students were
analyzed using implicative and cohesive statistics as an analytical tool. The results show
that it is important for students to understand the second derivative and the intervals into
which the domain is subdivided. The use of the particular statistical tool was found to be
both pertinent and highly useful since the results that were obtained were similar to results
obtained in previous studies using qualitative analysis.
Trigueros, M., & Lage, A. (2006). An analysis of students’ ideas about transformations of
functions. In S. Alatorre, J. L. Cortina, M. Saiz, & A. Mendez (Eds.), Proceedings of the 28thannual meeting of the North American Chapter of the International Group for the Psychology ofMathematics Education (pp. 23–30). Merida, Mexico: Universidad Pedagogica Nacional.
This study contributes to researchers’ and instructors’ understanding of students’
difficulties with transformations of functions. Students were interviewed while solving
problems involving such transformations. The results, which were analyzed using APOS
Theory, show that few students can work confidently with these problems involving
transformations of functions. The analysis showed limited evidence of students who had
interiorized the Actions involved in transformations of functions into Processes or who had
encapsulated those Processes into Objects.
222 12 Annotated Bibliography
Trigueros, M., & Martınez-Planell, R. (2010). Geometrical representations in the learning of
two-variable functions. Educational Studies in Mathematics, 73, 3–19.
This study is part of a project concerned with the analysis of how students work with
two-variable functions, a topic of fundamental importance in mathematics and its
applications. The authors investigate the relationship between students’ notion of subsets
of Cartesian three-dimensional space and the understanding of graphs of two-variable
functions. APOS Theory and Duval’s theory of semiotic representations are used in the
analysis. Nine students who had taken a multivariable calculus course were interviewed.
Results show that students’ understanding can be related to the structure of their Schema for
R3 and to their flexibility in the use of different representations.
Trigueros, M., & Oktac, A. (2005). La theorie APOS et l’enseignement de l’Algebre Lineaire.
Annales de Didactique et de Sciences Cognitives. Revue internationale de didactique des mathematiques (Vol. 10, pp. 157–176). IREM de Strasbourg, Universite Louis Pasteur.
The authors use APOS Theory to describe the students’ mental constructions of linear
algebra concepts. Special emphasis is placed on the concept of vector spaces since it is one
of the fundamental concepts of linear algebra and constitutes the beginning of an introduc-
tory course.
Trigueros, M., Oktac, A., & Manzanero, L. (2007). Understanding of systems of equations in
linear algebra. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the 5th Congress of theEuropean Society for Research in Mathematics Education (pp. 2359–2368). Larnaca, Cyprus:
University of Cyprus.
In this study, six students who were taking a course based on APOS Theory were
interviewed at the beginning of the course and at the end of the course in order to study
the viability of a proposed genetic decomposition of the concept of linear systems of
equations. The study also focused on students’ difficulties, their reasoning patterns, and
the evolution of their development of Schema (as defined in APOS Theory). Results show
that the students’ progress depended strongly on development of their Schema for variable.
The data also showed that a course based on APOS Theory helps students in the develop-
ment of their systems of equations Schema.
12.7 V Through Z
Vidakovic, D. (1996). Learning the concept of inverse function. Journal of Computers in Mathe-matics and Science Teaching, 15, 295–318.
This report is a part of a study that was conducted with five individual students and five
groups of students who were assigned to work together in the first course of the experimen-
tal calculus classes at Purdue University during the fall of 1992. The goal of the study was
to “discover” how the concept of inverse function can be learned, and hence taught, as well
as to investigate the differences between group and individual mental constructions of that
particular concept. The research followed the APOS research paradigm. It used the data to
obtain a genetic decomposition of the concept. On the basis of the genetic decomposition,
an instructional treatment was proposed. The instructional treatment consisted of computer
activities designed to encourage students’ development of a Schema for inverse functions.
This instructional treatment has not yet been implemented.
12.7 V Through Z 223
Vidakovic, D. (1997). Learning the concept of inverse function in a group versus individual
environment. In E. Dubinsky, D. Mathews, & B. E. Reynolds (Eds.), Cooperative learningfor undergraduate mathematics. MAANotes 44 (pp. 175–196). Washington, DC: Mathematical
Association of America.
The study was conducted with five groups of students working together on learning
activities and five individuals working alone on the same tasks. The mathematical issue
was the concept of function, in particular inverse and composition. The author was
interested in knowledge about the mental structures that underlie the cognitive develop-
ment of these concepts and differences between group and individual learning. An APOS
analysis of the data resulted in the development of a genetic decomposition for the concept
of inverse function and a related instructional treatment. The author also discovered that
students’ mental constructions were similar, regardless of whether they worked in an
individual or collaborative setting, but that the quality and quantity of their learning was
enhanced by working in collaboration with others.
Vidakovic, D., & Martin, W. O. (2004). Small-group searches for mathematical proofs and
individual reconstructions of mathematical concepts. Journal of Mathematical Behavior, 23,465–492.
The authors investigate and report on cooperative learning situations, specifically, how
individual ideas develop in a social context. Students with experience doing proofs in group
situations were videotaped working collaboratively on three mathematical statements.
Later, the students viewed segments of the group video and reflected on the activity of
their group. The authors observed changes in understanding that may have resulted from
parallel and successive interiorization and externalization of ideas by individuals in a social
context.
Vizcaıno, O. (2004). Evaluacion del aprendizaje del calculo desde una perspectiveconstructivista. Doctorate Thesis, CICATA, Instituto, Politecnico Nacional, Mexico.
This thesis is based on the idea that evaluation of the process of teaching and learning is
very important but also very complex and difficult to do. Moreover, simplistic methods can
even be counterproductive. Traditional methods of evaluation, which assign a number to a
student’s ability, can provide a distorted picture of the learning that may or may not have
taken place or is about to take place. As a result, such methods of evaluation may not
provide a reliable guide to improving pedagogy and its results in terms of learning. An
alternative to traditional methods of evaluation is the method of interviewing students
individually. Unfortunately, this requires more time and energy than is available to most
teachers.In this thesis, the author tests a third method based on APOS Theory and the ACE
Teaching Cycle. In this approach, evaluation is done through a series of instruments that
can be designed, administered, and scored within the normal progress of the class. To test
this method, the author used APOS and the ACE Teaching Cycle to teach a course in
calculus. The final grades (on a scale of 0 to 100) of the students were determined using the
evaluation methods proposed in APOS/ACE. Then an interview was conducted with each
student and the final grade was determined (using the same scale) again. Thus, the
comparison was between an evaluation method that was practical to use and a method
that was considered very accurate but highly impractical to use. The correlation between
the two sets of grades was very high (about 0.87), suggesting that the practical method
could be used to obtain high quality evaluations, at least in the case when the teaching was
based on APOS/ACE.
224 12 Annotated Bibliography
Weller, K., Arnon, I., & Dubinsky, E. (2009). Preservice teachers’ understanding of the relation
between a fraction or integer and its decimal expansion. Canadian Journal of Science,Mathematics, and Technology Education, 9, 5–28.
This article reports on the mathematical performance of preservice elementary and middle
school teachers who completed a specially designed unit on repeating decimals, including
the relation between 0:9 and 1. The teaching sequence was based on APOS Theory and
implemented the use of the ACE Teaching Cycle. The quantitative results suggest that the
students who received the experimental instruction made considerable progress in their
development of an understanding of the equality between 0:9 and 1 as well as between any
rational number and its decimal expansion(s). Students in a control group, who received
traditional treatment on these topics, made substantially less progress. For further analysis
of this study, see Weller et al. (2011) and Dubinsky et al. (2013) in this bibliography.
Weller, K., Arnon, I., & Dubinsky, E. (2011). Preservice teachers’ understanding of the relation
between a fraction or integer and its decimal expansion: Strength and stability of belief.
Canadian Journal of Science, Mathematics, and Technology Education, 11, 129–159.
In an earlier study of preservice elementary and middle school teachers’ beliefs about
repeating decimals, the same authors reported on a comparison of the mathematical
performance of 77 preservice teachers who completed an APOS-based instructional unit
with 127 preservice teachers who completed traditional instruction. The study was based on
interviews conducted 4 months after the instruction with 47 of these students. The
interviews revealed that the students who received the APOS-based instruction developed
stronger and more stable beliefs (over time) regarding their beliefs about repeating
decimals and the connection between repeating decimals and other rational number
representations. In their analysis, the authors develop a number of indices and categories
that may prove useful in other comparative studies involving interview and questionnaire
data with a large number of students. For further analysis of this study, see Weller et al.
(2009) and Dubinsky et al. (2013) in this bibliography.
Weller, K., Brown, A., Dubinsky, E., McDonald, M., & Stenger, C. (2004). Intimations of infinity.
Notices of the AMS, 51, 741–750.
The purpose of this article is to show how APOS, being a theory about how people come to
understand mathematics, can be helpful in understanding the thinking of both novices and
practitioners as they grapple with the notion of infinity.
Weller, K., Clark, J. M., Dubinsky, E., Loch, S., McDonald, M. A., & Merkovsky, R. (2003).
Student performance and attitudes in courses based on APOS theory and the ACE teaching
cycle. In Research in Collegiate mathematics education V. CBMS issues in mathematics
education (Vol. 12, pp. 97–131). Providence, RI: American Mathematical Society.
The authors examine the effectiveness of instruction based on APOS Theory and the ACE
Teaching Cycle using data from 14 previous studies in the areas of calculus, abstract
algebra, concept of function, quantification, induction, and the affective domain. The
results suggest that instruction based on APOS Theory may be an effective tool in helping
students to learn mathematical concepts.
Weller, K., Montgomery, A., Clark, J., Cottrill, J., Trigueros, M., Arnon, I., & Dubinsky,
E. (2002). Learning Linear Algebra with ISETL. Available from http://homepages.
ohiodominican.edu/~cottrilj/datastore/linear-alg/LLAWI-P3.pdf. Accessed 9 Jan 2013.
This is a first course in linear algebra employing the ACE Teaching Cycle using ISETL
activities. A matrix environment is implemented in ISETL to allow work in matrices over
finite fields and matrix algebra. The text covers vector spaces, solutions to systems of
12.7 V Through Z 225
equations, basis of vector space, and linear transformations. Extended topics include
matrices of transformations, change of basis, diagonalization, and eigenvectors/
eigenvalues. The text has 23 sections that allow for customization in a one-semester course
or the possibility of a two-semester sequence.
Zazkis, R., & Campbell, S. (1996). Divisibility and multiplicative structure of natural numbers:
Preservice teachers’ understanding. Journal for Research in Mathematics Education, 27,540–563.
This study contributes to a growing body of research on teachers’ content knowledge in
mathematics. The domain under investigation was elementary number theory. The main
focus concerned the concept of divisibility and its relation to division, multiplication, prime
and composite numbers, factorization, divisibility rules, and prime decomposition. The
APO (Action-Process-Object) framework was used for analyzing and interpreting data
acquired in clinical interviews with preservice teachers. Participants’ responses to
questions and tasks indicated pervasive dispositions toward procedural attachments, even
when some degree of conceptual understanding was evident. The results of this study
provide a preliminary overview of cognitive structures in elementary number theory.
Zazkis, R., & Gunn, C. (1997). Sets, subsets and the empty set: Students’ constructions and
mathematical conventions. Journal of Computers in Mathematics and Science Teaching, 16,133–169.
This study investigates preservice elementary school teachers’ understandings of basic
concepts of set theory. The students’ understandings are analyzed after instruction using
ISETL. Analysis of the data, which is based on APOS (not yet so called), reveals students’
difficulties with the idea of a set as an element of a set and the idea of the empty set.
Zazkis, R., & Khoury, H. (1994). To the right of the “decimal” point: Preservice teachers’ concepts
of place value and multidigit structures. In Research in Collegiate mathematics education I.CBMS issues in mathematics education (Vol. 4, pp. 195–224). Providence, RI: American
Mathematical Society.
The focus of this work is on preservice elementary teachers’ understanding of concepts
related to place value through the lens of the APOS framework. Special emphasis is put on
the de-encapsulation mechanism through a conversion task used in an interview. A genetic
decomposition for the construction of non-decimal number is presented.
226 12 Annotated Bibliography
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About the Authors
Ilana Arnon is a lecturer of Mathematics Education for prospective middle school
mathematics teachers at Givat Washington Academic College of Education, Israel.
Jim Cottrill is an assistant professor of Mathematics at Ohio Dominican University,
OH, USA.
Ed Dubinsky is a Visiting Adjunct Professor at the University of Miami, FL, USA.
Dr. Dubinsky is considered the father of APOS Theory.
Asuman Oktac is a professor in the Department of Mathematics Education at
CINVESTAV-IPN, Mexico.
Dora Solange Roa is an associate professor in the School of Mathematics at the
Universidad Industrial de Santander, Colombia.
Marıa Trigueros is a professor in the Department of Mathematics at Instituto
Tecnologico Autonomo de Mexico, Mexico.
Kirk Weller is a professor and head of the Mathematics Department at Ferris State
University, MI, USA.
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6,© Springer Science+Business Media New York 2014
233
Index
A
Abelian, 74
Abilities, 124, 130
Abstract
algebra, 2, 67, 68, 101, 106, 107, 178
objects, 3
Access, 113, 124, 129
Accommodation, 176, 182
accounts, 112
and assimilation, 19
Piaget’s theory, 113
re-equilibration, 122
ACE cycle, 101, 104, 105
ACE teaching cycle
APOS theory, 67
comparative analysis, 90
description, 58–59
first iteration
activities, 79–81
class discussion, 81
exercises, 81
repeating decimal, 78
genetic decomposition, 77
individual’s rational number
schema, 78
pedagogical strategy, 58
potential and actual infinity, 76
preservice teachers, 91
rational numbers, 75
repeating decimals, 76, 77
second iteration
activities, 82–84
class discussion, 84
encapsulations, infinite digit
strings, 82
exercises, 84–85
students perform actions, 77–78
third iteration
activities, 85–89
class discussion, 89
exercises, 89
infinite decimal strings, 85
supported encapsulation, 85
Achilles and Tortoise, 104
Across, 125, 126, 130–132
Action, Process, Object, and Schema (APOS)
theory, 1–4, 146, 154
ACE teaching cycle (see ACE teaching
cycle)
cognition and instruction, 179
coherence, 185
construction, mental structures, 181
design
and implementation, 57
instruction, 178
gauge students’ mathematical
performance, 91
genetic decompositions, 27, 35, 37–40
ISETL (see Interactive set theoreticlanguage (ISETL))
learning and teaching
abstract algebra, 67
binary operation, 70–72
class discussion, 73–74
exercises, 74
genetic decomposition, group schema, 67
group schema, 72–73
instruction results, 75
researchers’ experiences, 27–28
sets, 68–70
mathematical concepts, 181
mathematical knowledge construction,
17–26
mental structures, 29, 175
I. Arnon et al., APOS Theory: A Framework for Research and CurriculumDevelopment in Mathematics Education, DOI 10.1007/978-1-4614-7966-6,© Springer Science+Business Media New York 2014
235
Action (cont.)potential and actual infinity, 34
representations and transitions, 180, 181
teaching sequence, 51
Action-process-objects in APOS
conception, fraction, 163
interiorization, 152, 158, 173
postsecondary context, 153
Actions, 1, 3
abstract algebra, 101
ACE cycle, 78
activities and exercises, 94
APOS theory, 17, 175, 177
binary operation, 70–71
in calculus, 20
conception, 20, 66, 179, 186
construction
mental/physical objects, 19
mental structures, 18
existing physical, 66
external, 19
identification, quotient group, 102
interiorized, 76–77
in linear algebra, 20
mental process, 175
physical and mental transformations, 175
and process, 21, 22, 25, 26, 66–67
reconstruction, 176
set formation, 68–70
in statistics, 20
step-by-step, 19
students construct, 66
writing programs, 77
Activities, 2, 111, 114, 121
computer, 45
list, 52
teaching, 37, 40, 47–51
Activities, Class discussions, Exercises (ACE)
Teaching Cycle, 2
Actual infinity, 76, 104
Addition, 7, 9
AddString, 82, 83, 86–87
Ad infinitum, 183
Advanced mathematical thinking, 177
Affective domain, 107
Algebraic representations, 113
Analysis
data, 28, 93–94
empirical, 42
genetic decomposition, 38
ε–δ intervals, 45
interview data, 34
preliminary, 44, 51
text/instructional materials, 33
theoretical, 29
types, 48
Analytic, 113, 124, 125
Analytical, 1
Analytic geometry, 113
APOS-based instruction, 2, 57, 78, 90, 91
ACE cycle, 101
construction, mental structures, 179
design, 178
student attitudes, 107
APOS-based research, 2, 4
data collection and analysis phase, 95
semi-structured interviews, 96
A!P!O!S progression
developmental, 10
heart, APOS theory, 9
interiorization and encapsulation, 7
Approaches
closer and closer, 53
coordination, 47
didactical, 34
function, 45
students’ learning, two-variable
functions, 34
Approximation, 45
Aristotle, 104
Arithmetic, 147
Arithmetic reflection, 6
As linear independence, 107
Aspect, 110, 118, 120, 132
Assimilation, 113, 122, 176
and accommodation, 19
dynamic structure, 25
Attitudes
APOS theory, 104
student vs. APOS-based instruction, 107
Authors in APOS theory
Arnon, I., 3, 14, 137, 139, 140, 151–157,
159, 161–171, 173, 191, 197, 204,
210, 220, 224, 225
Artigue, M., 198
Asiala, M., 10, 12, 14, 19, 38, 65, 94, 96,
101, 102, 104–106, 110, 128, 163,
173, 191, 198, 199, 210
Ayers, T., 23, 148, 199, 227
Azcarate, C., 199
Badillo, E., 199
Baker, B., 14, 114–116, 118, 123, 125–127,
190, 199, 200, 203
Barbosa Alvarenga, K., 200
Baxter, N., 11, 200
Bayazit, I., 200, 201
236 Index
Bodı, S.D., 201
Boigues, F., 214
Breidenbach, D., 1, 20, 140, 201, 202
Brown, A., 14, 15, 34, 67, 68, 75, 140,
149, 184, 198, 201, 202, 211,
214, 225
Campbell, S., 202, 225
Campero, J., 222
Carlson, M., 202, 207
Cetin, I., 202
Clark, J.M., 14, 21, 22, 110, 113, 115,
116, 119, 128, 190, 203, 204, 214,
215, 225
Cooley, L., 15, 114, 123, 125, 129, 131,
133, 134, 190, 199, 203, 222
Cordero, F., 203
Cottrill, J., 13, 14, 44, 45, 47, 78, 99, 100,
113, 115, 116, 118, 119, 135, 190,
191, 198, 203, 204, 225
Czarnocha, B., 14, 128, 203, 204
Dautermann, J., 59, 210
Davis, G., 199, 204
DeVries, D., 198, 201, 203, 204
Dubinsky, E., 1, 5–15, 18–21, 23–26,
29–34, 45, 65, 67, 68, 75–77, 90,
97, 101, 103, 104, 110, 114, 118,
137–140, 144, 146, 148, 149, 151,
155, 162, 180, 185, 191, 192,
197–201, 203–213, 216, 219, 220,
223–225
Elterman, F., 207, 210
Ely, R., 211
Escandon, C., 222
Estruch, V.D., 214
Fenton, W.E., 212, 219
Font, V., 199, 212
Garcıa, M.M., 12, 17, 25, 112–114, 128,
190, 192, 203, 212, 217, 218
Gavilan, J.M., 212
Gimenez, J., 212
Gong, C., 207, 210
Gray, E., 15, 201
Gunn, C., 226
Hahkioniemi, M., 212
Hamdan, M., 170, 213
Harel, G., 208, 213
Hawks, J., 201
Hemenway, C., 200, 203
Hernandez Rebollar, L.A., 213
Hollebrands, K.F., 213
Kabael, T., 214
Khoury, H., 226
Kiaie, C.C., 219
Kleiman, J., 198
Ku, D., 15, 35–38, 102, 214
Kuhn, J., 2, 93, 220
Lage, A., 222
Leron, U., 67, 68, 75, 201, 208, 210
Levin, G., 11, 200
Lewin, P., 31, 199, 206, 208, 210
Llinares Ciscar, S., 201, 212, 214
Loch, S., 225
Lozano, D., 218
Malaspina, U., 212
Mamolo, A., 215, 220
Manzanero, L., 223
Martınez-Planell, R., 22, 34, 106, 215, 222
Martin, W.O., 224
Mathews, D., 21, 22, 198, 199, 209, 214,
215, 223
McCabe, G.P., 220
McDonald, M.A., 14, 18, 110, 113, 114,
118, 197, 201, 208, 211, 215, 225
Meel, D., 215
Mena, A., 216
Merkovsky, R., 225
Miranda, E., 203
Moreira, R.N., 170, 216
Morics, S., 199
Nesher, P., 152, 197
Nichols, D., 201, 204
Nirenburg, R., 197, 198
Oktac, A., 15, 28, 41, 42, 44, 47, 103, 106,
111, 199, 214, 218–220, 223
Parraguez, M., 15, 111, 218
Pegg, J., 218
Possani, E., 218
Prabhu, V., 204
Preciado, J.G., 218
Przybylski, J., 219
Ramirez, A.A., 219
Reynolds, B.E., 219, 223
Roa-Fuentes, S., 15, 28, 41, 42, 44, 103,
106, 219, 220
Salgado, H., 220
Schwingendorf, K.E., 198, 204, 209,
219, 220
Sinclair, N., 220
Stenger, C., 15, 25, 108, 211, 220, 225
St. John, D., 203
Strobel, K., 215
Tabaghi, S.G., 220
Tall, D., 15, 75, 170, 204, 206, 218
Thomas, K., 198, 201, 202, 204, 221
Tolias, G., 202, 203
Tossavainen, T., 221
Index 237
Authors in APOS theory (cont.)Trigueros, M., 15, 22, 34, 36, 47, 106,
112, 199, 200, 203, 213–215, 218,
220–223, 225
Vakil, R., 203
Valls, J., 201
Vidakovic, D., 107, 203, 204, 211, 220,
223, 224
Vizcaıno, O., 224
Weller, K., 1, 14, 15, 29, 34, 48, 59, 76, 77,
90, 103–105, 107, 108, 137–141,
149, 177, 178, 201, 210, 211, 220,
224, 225
Wilhelmi, M.R., 212
Wilson, R.T., 181, 210
Wodewotzki, M.L.L., 216
Yiparaki, O., 210
Zazkis, R., 141, 202, 210, 215, 225, 226
Auxiliary questions, 107
Awareness, 124, 125, 130–131
Axiom, 66, 67, 72, 73
Boolean-valued function, 49
funcs, 51satisfied and false, 51
schema, 48–49, 67, 68, 72, 75
set and binary operation, 48
Axis, 126
B
Bases, 111
Basis
concept, 35
concept image, 12
linear algebra concepts, 15
R3, 54
and spanning set, 35
spanning sets, 37
of vector space, 103
working hypotheses, 27
Behavior of a curve, 115
Binary operations, 14, 96, 103
activities, students, 74
addition and multiplication, 50
axiom Schema, 75
definition, 49
encapsulation, 185
funcs, 70input to, 30
ISETL, 70–72
mental Object, 176
pair, 72–74
schemas, 67, 68, 71, 72
variables, 63–64
“vector addition”, 43
vector space, 48
Boolean operators (as defined and used
in ISETL), 62, 83
Boolean-valued function, 49, 51
C
Calculus
actions, 20
and analysis courses, 135
differential and integral, 112
encapsulation and objects, 22
instruction and performance, 178
interiorization and processes, 21
mathematical concepts, 12, 18
reflective abstraction and reports, 13
Calculus graphing problem, 123, 129
Calculus graphing schema
development, interval stage, 124–128
performing Actions, 131
property stage, 124
students’, 123–124
thematization, 135
Cardinality
infinite and finite sets, 182
process, 49
tennis ball problem, 184
Cartesian plane schema, 111
Cartesian product, 48, 49
Chain rule
interview, 135
intra-stage, 115–116
mathematical concepts, 14
schema, 116–118
students, 113
trans-stage, 119–121
Change
Clara’s thematized schema, 134
derivative, 116, 118
inter-stage, 120
intra-stage, 120
schema, 109
Circle, 53
Class, 109, 110
Class discussion, 2
Classification and seriation, 8, 9
Classification of research studies
comparative studies, 104, 105
level of cognitive development, 104, 106
non-comparative studies, 104, 105
student attitudes, 104, 107
238 Index
Class inclusion, 110
Class plan, 52
Classroom observation
APOS theory, 102
methodological design, 95
CLUME. See Cooperative Learning in
Undergraduate Mathematics
Education (CLUME)
Cognition
APOS theory, 178
construction, 182
description, 33
and epistemology, 28
genetic decomposition, 29
mathematical concept, 179
Cognitive
construction, 112
developments, 8, 9
lower, 6
progression, 138
structure, 6
Coherence
notion, 13
of schema, 25, 73, 78, 112, 114, 124, 125
trans-stage development, 118
Coherent
APOS theory description, 14
framework, 110
object collections, 12
schema, 111, 131
Collaborative research, 95
Collection of data
classroom observations, 102–103
historical/epistemological analysis,
103–104
interviews, 95–100
textbook analyses, 103
written questions, 100–102
Combinations of functions, 36, 111, 198, 213
Combinatorics, 11
Commutativity
addition and property, 7
and Piaget’s phrase, 8
Compactness, 12
Comparative analysis, 105
Comparative studies, 104, 105
Comparison, 8
Completed infinite process, 76
Completion of the process, 182–183
Complexity, 112
Components
complexity level, 112
individual, 112, 128
of research, 94
schema, 110–112, 129, 132
Composition, 70, 72–73, 113–114, 116, 119
Computer activities
computer algebra system, 103
students’ thinking, 11
types, 45
Computer algebra system, 103
Computer games, 95
Computer laboratory, 171
Computer procedure, 138
Computer programming
pedagogical tool, 11
process, object and spoke, 10
writing, debugging and running, 10
Computer science, 1, 179
Concavity, 126, 127, 134–135
Concept
abstract algebra, 106
action, 19, 102
actual and potential infinity, 149
APOS theory, 107, 108, 146
cognitive developments, 140
and conception, 18
cosets, normality and quotient groups, 101
emerging Totality, 99
equivalence classes, fractions, 111
historical developments, 140
image, 12–13
learning, 138
linear transformation, 103
mathematical, 17–19, 25, 103–104, 110,
122, 150
n-tuple, 20preliminary genetic decomposition, 94
process to object, 138
schemas, 109, 122, 135
textbooks, 103
triad stages, 121
vector space, 111
Conception
action, 20
and concept, 18
object, 2, 22
process, 22
students, 22
ε–δ Conception, 100Concrete and abstract
APOS theory, 153, 154, 164
circle cutouts, 152
Concrete manipulatives, 185
Concrete objects, 3
Concrete operations, 3, 138
Index 239
Conjunction, 31
Connection, 112, 113, 125
Conscious, 128, 130–131, 135
Consensus, 95, 96
Constructivism, 1
Constructivist, 1
Constructs
linear transformation concept, 41
mental, 36–37
prerequisite, 36
process and object conceptions, 44
Content
and operations, 6, 7
profound effect, 10–11
SETL, 11
Context
mathematical concepts, 181
process conception, 176, 181
Contiguous, 124, 125
Continuity, 123, 129, 132–134
Control group, 105, 224
Cooperative, 2
Cooperative learning, 94, 107, 179–180
Cooperative Learning in Undergraduate
Mathematics Education (CLUME),
13, 14
Cooperatively, 2
Coordinated schema, 100, 106
Coordination, 110, 118, 124, 130, 175, 186
axiom schema, 72
child realizes, 9
construction, 75
de-encapsulation and reversal process,
22–24
description, 9
individual, 68
interiorization, 17, 20
mental mechanisms, 5
mental structures, 1, 10, 12
set and binary operation, 71, 73
1–1 Correspondence, 8–9
Correspondences, 112, 113
Cosets
action conception, 102
chain rule, 114
concepts, students’ learning, 101
formation, ability, 102
geometric representations, 187
operations, 177
Counting numbers, 182, 183
Course, 124, 132, 135
Criteria, 144–148
Criterion, 116, 119
Curriculum development, 1, 2, 4
Curves, 111, 115
Cusp, 127, 135
Cycle
ACE teaching, 101, 104, 105
research and curriculum development,
93–94
D
Data
analysis, 123, 129
classroom observations, 102
collection (see Data collection)comparative, 105
“off-line”, 13
and report, 13, 14
unanalyzed, 13
Data analysis
classroom observations, 102–103
historical/epistemological analysis, 103–104
interviews, 95–100
refinement cycle, 29
revisions, 45
textbook analyses, 103
written questions, 100–102
Data collection
classroom observations, 102–103
historical/epistemological analysis, 103–104
interviews, 95–100
textbook analyses, 103
written questions, 100–102
Data type, 59, 64–65
Dec2Frac, 83–84, 86–87
Decimal expansion, 75, 77, 81–83, 87, 105
Decimal expansion package, 78, 82, 83, 85, 89
Decreasing, 126, 127, 132, 134
De-encapsulation
binary operation Object, 176
coordination and reversal process, 22–24
mental structures, 1
Definite integral, 14
Definition, 118, 120, 122, 132
Definition of limit, 100
Derivative
chain rule, 116, 135
genetic decomposition, 53–54
graduate student in seminar, 52
graph, 14, 113–114
inter-stage, 116
intra-stage, 114
schema, 122
trans-stage, 118
240 Index
Design
classroom observation, 102–103
genetic decomposition
historical development, concept, 34
mental constructions, 36–37
preliminary, 33
prerequisite constructions, 36
spanning set and span, 35–36
teaching activities, 47–51
instruction, 93–94
interview questions, 95–96
written questions, 101
Design of instruction, 93–94
Design of interview questions, 95–96
Determinants, 11
De-thematizing, 110, 128
Development, 1–3
activities designed to facilitate, 49–21
binary operation, 48
mental, 30–31
preliminary genetic decomposition, 28–29
refinement, 47
spanning set and span concepts, 36
students’ constructions, 28
student’s function schema, 32–33
theories/models, 27
Development of a schema
description, 112–113
inter-stage, 116–118
intra-stage, 114–116
Piaget’s theory, 113
trans-interval, trans-property level, 131
trans-stage, 118–121
triad, 113
Didactical route, 96
Differentiate, 122, 134
Difficulties, 110, 123, 126, 127
APOS theory, 107
encapsulation, 147
interview questions, 95
mathematical infinity, 103
0.999, mental object, 99
process to object, 140
widespread, 140
Dimension, 37, 111, 118
n-Dimensional spaces, 111
Discrepancies, 38
Discrete mathematics, 11
Disjunction, 31
DivString, 82, 83, 88
Domain
function, 41, 125
graph, 131
individual’s function Schema, 122
intervals, 125, 127, 130
positive integers, 32
process, 45–47
and range, 30, 32, 52
re-equilibration, 122
schema, 118
sets, 124
transformation types, 30
vectors, 42, 43
Dynamic conception
coordination, two process, 47
static conception, 45
Dynamic structure, 21, 25
E
Element
four, 9
transform, 6
Elementary school
APOS theory, 152–154,
173–174
equivalence classes, fractions,
173
learning, fractions, 161
mathematics learning, 3
postsecondary students, 151
Emerging
object, 147
process, 145
totality, 145
Emerging totality (ET), 99
Empirical
analysis, 42, 44
evidence, 94, 95, 103
studies, 29, 40–41
Encapsulation, 1, 94, 100, 102
ACE cycle, 78, 82
action, 66, 77
axioms, 68
binary operation, 71, 185
infinite string, 77
input/output, program, 67
mechanism, 25
mental
mechanism, 76
object, 69, 70
structures, 175
and objects, 21–22
reflective abstraction, 18
Entry, 97–99
Epistemological study, 95
Index 241
Epistemology
genetic decomposition, 181
inextricably interwoven, 12
mathematical concepts, 181
Equality 0.9 ¼1
ACE teaching cycle, 105
mental Object, 99
EqualString, 83
Equation, 142, 147
linear function, 53
and matrices, 49
plane/line, 54
solution set, 36
solving systems, 35–36
Equilibration, 113
Equivalence classes of fractions
concept, 111
definition, 174
grade 5, 169
process conception, 152
teaching experiment, 170
Errors, 27, 51, 52
Euclidean geometry, 113
Evidence, 125–126, 131, 133, 135
APOS-based research, 101
cosets, 102
student data, 103
Exams
interview subjects, 95
non-comparative data, 105
written questions, 100–101
Exercises, 2, 94–96
Experimental course
APOS-based, 107
non-comparative data, 105
Experimental group, 223
Exponential functions, 116, 119
Exponential rule, 115, 119
Expression
process conception, 141
totality, 147
External cue, 102
Exxon Educational Foundation, 14
F
Figures, 113, 130
Finite
cardinality, 182
decimals, 76
encapsulation, 176, 184
enumeration, 182, 183
field, 50
number, 144
process, 137, 150
First derivative, 126, 127, 130
Fixed, 133
Flexible, 127, 130–131, 135
Flexibly, 133
Follow-up questions, 96
Forever, 141–145, 147
for loop, 62
Formal definition of limit, 100
Formal thinking, 45
Frac2Dec, 83–88
Fractions
activities, students, 86, 87
arithmetic operations, 158
binary operations, 158
circle cutouts, 152, 153, 171
commands, 87
construction, 159
decimal strings, 84
encapsulation, 163
equivalence classes, 151, 170
Frac2Dec, 83
individual strings, 85
infinite decimal strings, 85
ISETL, 85
K–12 level, 178
and non-numeric ratios, 186
part-whole interpretation, 152,
154, 155
relation, 185
repeating decimal, 77, 84, 89, 90
study, 3
Framework, 110
Framework for analysis (FFA)
interpretations, 143
progression, 145, 148
funcs, 62–65, 69–73, 79, 83axiom facilitates encapsulation, 51
ISETL command, 50
tests, 51
tuple addition and scalar multiplication, 50
Functional
analysis, 6
derivative, 114, 120, 135
exponential or trigonometric, 116
graph, 113–114, 123, 129
individual’s Schema, 122
inter-stage, 116, 124
intra-stage, 114, 124
schema, 111, 112
trans-stage, 118, 124, 125
triad, 113
242 Index
Functions
abstract objects, 3
action and process conception, 9, 179
APOS theory, 107, 177
child’s culture, aptitudes and interests, 8
composition, 23
concept, 19, 21
genetic decomposition, 29–30
independently, 1
linear transformation, 103
and logic Schemas, 30
mathematical concepts, 12
process, 23
propositional and predicate calculus, 11
proposition-valued function, 32
representations and transitions, 180
schema, 111–113, 122
schema for R3, 106
space, 6
of two variables, 106
Fundamental planes, 106
G
Generalization, 17–19, 175
General transformation concept, 103
Generic student, 29
Genetic decomposition, 2, 11, 123–125, 127
ACE cycle, 58
Action toward Object, 90
activities phase, 59
APOS theory, 182
central role in APOS-based research, 37–40
common error, 51–54
constructing process and object
conceptions, 41–44
data collection and analysis phase, 95
design
mental constructions, 36–37
prerequisite constructions, 36
spanning set and span, 35–36
epistemological analysis, 181
framed, 17
function, 29–30
hypothesized theoretically and tested
empirically, 17–18
induction, 30–32
infinite repeating decimals, 76, 77
learning process, 18
for limit, 99, 100
linear transformation, 103
mathematical object, 178
mental constructions, 58, 91, 94
mental structures, 177
prerequisites, 41
refinement, 44–47, 99
representations and transitions, 180
schema, 67
students’ learning, 101
teaching activity design
facilitate development, 49–51
vector space, 48–49
Genetic epistemology, 113, 181
Geometric figures, 113
Geometric structures, 112–113
Global behavior, 130–131
Global entity, 113
Graphical, 124, 125
Graphing
and derivative, 14
schema, 125–126, 131
Graphs
actions, 125, 126
and algebraic expressions, 53
continuity condition, 132
function, 123
process, 124
relations, 11
schema, 125–126
Stacy’s sketch, 127
transformed function, 52
translations, 52
GreatString, 83
Groups
annual meeting, 14
APOS, 57, 107
chain rule, 119
concept/property, 7
cosets, normality and quotient, 14
interview questions, 95
procedural and conceptual
understanding, 90
quotient, 105, 106
schema, 72–74, 114
teaching and learning (see Teachingand learning groups)
work, 107
written questions, 100–102
Grundvorstellungen, 182
H
Hilbert’s Hotel, 104
Hint, 96
Historical development, 112–113
Historical/epistemological study, 95
Index 243
Historical study, 95
Homework exercises, 95–96
Hypotheses, 27, 37
I
if statement, 62
Image
mental, 31
spanning set, 35
vectors, 43
Imagery
APOS, 154
authentic anticipatory, 166
concrete objects, 164–169
Imagination, 3, 138, 149
Implementation of instruction
APOS-based research and/curriculum
development project, 93
collection and analysis of data, 94
genetic decomposition, 103
Implications
actions, 31
implication-valued function, 32
process, 32
Implicit
differentiation, 115
functions, 116, 120
Incomplete process, 76
Increasing, 114, 126, 127, 132
Induction
genetic decomposition, 30–33
mathematical, 33
Inequalities, 100
Infinite
APOS-based research, 139
cardinality, 182
FFA, 144
iterative processes, 183
mental construction, 176
object, 140
repeat, 2
repeating decimals, 176, 178
totality and levels, 137
Infinite decimal string (decimal string)
class discussion, 81, 84
exercise, 81, 89
MultString and DivString, 88
process conception, 81
students, 84
Infinite repeating decimal
conceptual grasp, 90
encapsulation, 87
equality 0.9¼1, 99
genetic decomposition, 76, 77
Infinite repeating decimals, 2, 137,
143, 150
Infinitesimals, 75–76, 104
Infinity, 15, 103–104
Infix notation, 63, 71–72
Informal, 45
Inputs, 111
Instruction
APOS-based approach, 41, 179
APOS-related approach, 105
classroom observations, 102
design, 40
implementation, 28
materials, 33
mathematical concept, 179
mathematical performance, 95
mental structures, 179
preliminary genetic decomposition, 106
reform-oriented approach, 179
research cycle, 47–48, 93–94
student attitudes, 107
theoretical analysis, 94
treatment, 45
Instructional design, 105
Instructional materials, 1
Instructional strategies, 179
Instructional treatment, 138, 142
ACE cycle, 77
APOS theory, 201, 202, 205
computer activities, 45
conversion techniques, 78
effectiveness, 199
genetic decomposition, 223–224
Instrument
and/or in-depth interviews, 28
interviews, 96
theory-based, 37–38
written questions, 101
Integer
corresponding strings, 86
decimal strings, 84
division, 8
exercises, students, 89
operation/process, forming units, 6
physical objects, 7
positive, 8, 9
repeating decimal, 77, 90
set, 74
view, 80
Integral, 112, 116, 119, 120
Integrate, 120, 126, 134
244 Index
Interaction of schemas
calculus graphing schema, 124–128
individual, 122
students’ calculus graphing schema,
123–124
Interactive Set Theoretic Language (ISETL)
description, 59
operations, data types, 64–65
pedagogical tool, 65–67
supporting mathematical features,
62–64
syntax resembles standard mathematical
notation, 59–61
Inter-calculus, 112
Inter-function, 112
Interiorization
action, mental process, 66
actions, 25, 102, 106
APOS theory, 156, 164
concrete objects, 152
constructed—interiorization, 1
digits, indeterminate length, 76–77
encapsulation, 154
ISETL, 70
mechanism, 17, 175
mental mechanisms, 77
mental objects and actions, 20
mental process, 175
non-unit fractions, 160–162
overall interiorization, 161
preliminary genetic decomposition, 99
and process, 20–21
quantified statement, 71
student, 68, 81
young child’s, 3
Internal relations, 113
Internal reviews, 14
Interpretation
analyzing student responses, 103
interviews, 96
Inter-stage, schema
chain rule, 116–118
derivative, 116
function, 116
interval, 124
processes and objects, 116
property, 124
Intervals
actions/processes, 114
domain, 123, 130
reconstruct and coordinate, 124
schema, 124–128
schemas (see Schemas)
silence, 96
smaller, 100
students’ abilities, 124
trans-stage, 130, 132
Interview analysis, 96
Interviewer, 96, 107
Interviews
analysis, 96
APOS-based research, 95, 96
college students, 13
conducted, 34
data, 34
extract, 97
genetic decomposition, 99
homework exercises, 95–96
linear algebra textbook, 39
mental constructions, 95
preliminary genetic decompositions, 106
protocol, 95
questions, 38
refinement, 99
research instruments, 101
transcription, 34
written instruments and/or
in-depth, 28
Intra-calculus, 112
Intra-figural stage, 113
Intra-function, 112
Intra-stage, schema
APOS theory, 114
chain rule, 115–116
derivative, 114–115
functions, 114
interval, 124
property, 124
Intuition
existing structure, 182
mental constructions, 182
principle, accumulation, 183, 184
Invariant, 130–132
Invertible, 44
ISETL. See Interactive Set TheoreticLanguage (ISETL)
Isolation, 114, 116, 123, 124
Issues
context, group theory, 96
interview segment, 99
mathematical development, 104
organization and analysis, 99
potential and actual infinity, 104
Iteration
finite sequence, 183
principle, accumulation, 183
Index 245
K
Knowledge
construction, 122, 127–128
development, 6, 113
structure, 110, 111
Kuhn, T.S., 93
L
Laboratory activities, 11
Lagrange’s theorem, 101
Learn
action, 106
APOS theory, 107
design activities, 47
genetic decomposition, 27, 28, 34
instructor/researcher, 94
mental constructions, 52
pedagogical strategies, 94
preliminary genetic decomposition, 106
social component, 107
Learning
APOS theory, 29
data analysis, 29
design, activities and teaching
sequences, 40
genetic decomposition, 35
knowledge, APOS theory, 27–28
linear algebra, 49
Lecturing, 94, 104
Lecturing instruction, 179
Leibnitz rule, 119
Lens, 38
LessString, 83
Levels, 2, 3, 99, 104, 106, 137–150
Levels between APOS Stages, 151
Limitations, 107–108
Limits, 14, 123, 127, 129
action, 53
APOS-based research, 107–108
applications, 45
concept, 44
function sequences, 30
genetic decomposition, 99
preliminary genetic decomposition, 46
process, 53
secant line, 54
starts, 45
Linear
algebra (see Linear algebra)dependence, 37
equations, 15
independence, 37, 54
string, 9
transformations, 15
Linear algebra
actions, 20
APOS theory, 178
concepts, 15
encapsulation and objects, 22
interiorization and process, 21
interview, 39
ISETL, 48
schema, 49
Linear combinations
basis vectors, 35
constructing process, 37
given vectors, 35
mental Object, 41
schema, 111
Linearity
properties, 41–42, 44
transformation (see Linear transformations)
Linear transformations
algebra concepts, 15
algebra textbooks, 103
construction, 41–42
exemplified, 34
genetic decompositions, 103
interviews, 106
object conceptions, 44
preliminary genetic decompositions, 41
study, 122
Links, 113
Logic
of actions, 110
connector, 44
“reflective abstraction”, 7
schemas, 30–32
Logical connector, 44
Logico-mathematical
construction, 6
structures, 6
Logico-mathematical structures, 110
M
Maps, 9
Material action
and interiorized operations, 7
transforming physical objects, 7
Mathematical concept
APOS theory, 17, 107, 181
been built, 25
cognition and instruction, 179
hierarchical manner, 19
246 Index
learning, 40
mental constructions and mechanisms, 17, 94
mental structures, 178, 181
preliminary genetic decomposition, 94
student observations, 33
transformation, 28
Mathematical induction, 10, 12, 178
Mathematical infinity, 76, 103, 178
Mathematical knowledge
actions, 19–20
de-encapsulation, coordination and
reversal, 22–24
encapsulation and objects, 21–22
interiorization and processes, 20–21
structures and mechanisms, 25–26
thematization and schemas, 24–25
Mathematical performance, 95, 100–101, 104
Mathematical problem situations, 8, 13
Mathematical programming language
genetic decomposition, 59
ISETL (see Interactive Set TheoreticLanguage (ISETL))
Mathematicians, 103, 104, 108
CLUME participants, 14
education research, 5
research, 6, 7
Mathematics education, 180–182
Matrices
actions to define size, 51
equations, 49
sets, 36, 74
textbook, 11
n-tuples and polynomials, 35
Mechanisms
encapsulation, 30, 34
and mental structures, 27–28
Mental constructions
APOS-based research, 101
binary operation, 48
development, 176
genetic decomposition, 28, 34, 36–37, 94, 178
learning, concept, 35
mathematical performance, 95
schemas, 49
spanning set and span, 36
theoretical analysis, 38, 95, 182
Mental mechanisms, 1, 3, 66, 77, 94, 175
actions, 19–20
APOS theory, 17
de-encapsulation, coordination and reversal
process, 22–24
description, 18
encapsulation and objects, 21–22
interiorization and processes, 20–21
thematization and schemas, 24–25
Mental object
actions, 10
encapsulation, 8, 12, 104
equality, 99
Mental structures, 1, 66, 77
APOS theory, 175
genetic decomposition, 177, 178
mathematical concept, 181
Mental structures/constructions, 17
Metacognition, 180–182
Metaphors, 180–182
Methodological design, 95
Methodology, 93, 95, 108
Middle school, 1, 2
Mind, 112–114
Misunderstandings, 2, 52
Models
APOS theory, 1, 2
construction, 34
definition, 27
epistemology and mathematical concept
cognition, 28
theoretical, 37–38, 40, 47
Modifications of the pedagogical strategies, 101
Modus ponens, 32
Multiple authors, 96
Multivariable functions, 122
Multivariate calculus, 106
MultString, 83, 88–89
N
National Science Foundation (NSF), 13
Natural numbers
construction, 184
encapsulation, 184
finite segments, 104
subset, 182
Negation, 31
Negative, 114, 126, 127
Non-Cartesian spaces, 111
Non-comparative studies, 104, 105
Nonlinear transformation, 41
Normality, 14
abstract algebra concepts, 106
learning, 101
Number
conference presentations and published
papers, 12
copies, 8
counting, 7
Index 247
Number (cont.)finite, 144
objects, 7
real, 122
and segment, 113
substantial, 138
units/uses, 9
O
Objects, 64, 67–69, 71, 84
abstract, 3
abstract algebra, 101
actions, 102
activities and exercises, 94
binary operation, 176
conception, 2, 68, 72, 179
coordination mechanism, 23
emerging Totality, 99
encapsulation, 184
and encapsulation, 21–22
genetic decomposition, 177–178
mathematical performance, 95
mental structures, 179
preliminary genetic decompositions, 103
transformation, 175, 177
Observation, 95, 102–103
Operational, 15
Operational definition, 144, 146, 148
Operations, 111, 114, 116, 128
arithmetic, 185–186
binary, 14
and content, 6, 7
encapsulation, 185
interiorized, 7, 8
mental Object, 176
and structural conceptions, 15
temporal constraints, 183
Orbits subgroups, 74
Order
and classification, 8
formation, 9
independent, 7
maintain consistency, 12
SETL, 11
Ordering, 6–9
Output, 11
P
Paradigm
APOS theory, 2, 93, 108
definition and characteristics, 93
Paradox, 104, 184
Parameter, 35, 36
Partial derivatives, 122
The Part-whole interpretation of fractions
action and process, 152
construction, 155
Pascal, 10
Pedagogical strategies
APOS-based research, 101–103
cooperative learning, 94
textbooks, 103
Pedagogical suggestions, 94
Pedagogy, 11–12, 179
Performance, 95, 96, 104, 124
students, 34, 38, 48
variations in mathematical, 28
Permutations (permutation group), 14, 70, 74
Perspectives, 113
Phenomena, 7
Physical objects
action, 8
transforming, 7
Piaget, J., 175, 177, 180–182, 185
Piaget’s stages of cognitive development, 152
Pilot interview, 95
Poincare, 104
Points, 111, 113, 118, 132
Polynomials, 35, 36
Positive, 114, 126
Positive integer, 8, 9
Postsecondary, 1, 3
Potential infinity, 76, 104
Power rule, 115, 116, 119
Predicate calculus
mathematical concepts, 12
and propositional, 11
Predict
constructions, 44, 47
and diagnostic tool, 38
genetic decomposition, 29
mathematical concept and conditions, 27
preliminary genetic decomposition, 38
Preliminary, 149
Preliminary genetic decomposition
design, 33
empirical, 34
instructional treatment development, 28
interiorization, step 2P, 99
interview, 106
linear transformation concept, 41, 106
mental constructions, 101
properties, 42
refinement, 2, 45, 46, 100
248 Index
single process, 47
textbooks, 103
Prerequisite
concepts, 36, 107
constructions, 35, 36, 41
structures, 28
Preservice elementary and middle teachers, 75,
76, 78
Principle of accumulation
APOS theory, 184
iteration, 183
mental construction, 184
Problem situation, 110, 112, 118, 128, 130
Problem solving, 94
proc, 63, 64Procept, 15
Process, 1–3
abstract algebra, 101
action, 66–67, 90
activities and exercises, 94
axiom, 68, 72
cognition, 178
conception, 78, 81, 176, 179, 180
de-encapsulation, coordination
and reversal, 22–24
encapsulation, 100, 178
enumeration, 182, 183
finite and infinite, 184
infinite string, 77
and interiorization, 20–21
inverses axioms, 73
mental
action, 76
reflection, 66
structures, 179
transformations, 175
object, 76, 90, 99
potential infinity, 104
preliminary genetic decompositions,
103, 106
reconstruction, 176
set, 68
totality, 90
vector space, 103
Product of cosets, 101
Programming, 103
Programming language, 138
Progress
dynamic conception hinders, 47
limited, 38
Progression, 112–114
APOS theory, 137
FFA, 144, 148
historical developments, 140
obstacles, 140
process and object conception, 138
totality, 149
Projective geometry, 113
Prompting, 96
Properties
actions, 125
coordinate, 127, 132
graph, 113–114
intervals, 131, 133
objects, 7
processes, 123, 125
recognition, 124
schema, 123, 124
set, 112
Proportion, 8
Propositional calculus, 11
Propositions
abstract objects, 3
first-order, 31
positive integers, 32
Proposition-valued function, 32–33
Q
Quantification
APOS theory, 178
conception, 45
domain approach, 45
schema, 45, 100
universal, 43
Questionnaire, 95, 100
Quotient groups, 14
APOS theory, 107
cosets and normality, 14
pedagogical approach, 105
students’ learning, 101
R
Range, 118, 122
and domain, 30, 32, 41
function change, 52
process, 45–47, 100
vectors, 43
Rate, 53
Rate of change, 115, 116, 118
Rational numbers, 67, 75–76, 78, 89
!rational off, 61!rational on, 61Real numbers, 122
Reasons, 109, 113, 125–126
Index 249
Reassemble, 133
Recognition, 116, 120, 124
Reconstruction
and reorganization, 6
schema, 122, 128–129
Re-equilibration, 122
Refined genetic decomposition, 38, 44, 99,
100, 106
Refinements
cycle, 29
genetic decompositions, 44–47
preliminary genetic decomposition, 2, 40
research data, 29
Reflection
APOS theory, 180
metacognition, 180
reflective abstraction, 6
Reflective abstraction, 1, 17, 18, 58, 65,
177, 180
Reform oriented approach to instruction, 179
Regions, 111, 122
Reinterpretation, 113
Relations
a/b and c/d, 8derivatives, 122
and graphs, 11
integers, 8
inter-stage, 114
intervals, 125
objects, 8, 112
schemas, 110, 111, 135
transformations, 114
Relation to theory, 98, 99
Removed, 131–134
Reorganization, 6
Repeating
FFA, 144
process stage, 144
Representations, 3, 113, 118, 124
actual infinity, 76
APOS theory, 180
Boolean-valued func, 73cosets, 187
fraction/integer, 84, 86, 89
genetic decomposition, group schema, 67
ISETL, 60, 87
notational scheme, 78
repeating decimal expansions, 85, 87
tennis ball problem, 183–184
transition, 180
Research
cognitive development level, 106
comparative studies, 105
and curriculum development, 93–94
cycle, 94
data collection and analysis
classroom observations, 102–103
historical/epistemological, 103–104
interviews, 95–100
textbook, 103
written questions, 100–102
development cycle, 93–94
instrument, 95, 101
non-comparative studies, 105
questions, 108
scope and limitations, 107–108
student attitudes and long-term impact, 107
Research in Undergraduate Mathematics
Education Community (RUMEC), 1
Reversal, 1, 5, 10, 12, 22–24
Revision of the genetic decomposition, 99
Rn, 122
Role, 110, 113, 120
Role of genetic decomposition
design, 47–51
hypothetical model, 27
research, 37–40
Rule of assignment, 42
S
Scalar
process, 42
vector addition and scalar multiplication,
42–43
vectors, 36
Scalar field, 36
Schemas, 1, 2
assimilation, new constructions, 122
axiom, 68, 75
binary operation, 68, 72
calculus graphing schema (see Calculusgraphing schema)
chain rule study, 135
construction, 178
n-dimensional spaces, 111
functions, 111
genetic decomposition, group, 67
group, 72–73
individuals, 111
individual’s rational number, 78
inter-stage, 116–118
intra-stage, 114–116
mental structures, 175
mind, individual, 112–114
Piaget’s work, 109–110
250 Index
structure, 110
thematization (see Thematization)
and thematization, 24–25
trans-stage, 118–121
Scope, 107–108
Scope of a schema, 110, 114, 118
Script comment, 98–99
Scripted interview, 97
Scripting of an interview transcript, 97
Secant
action, 53
paragraph, 53
schemas, 53, 54
Secondary school, 1, 3
Second derivative, 113–114, 123, 129
Segment
interview transcript, 144–146, 148
and number, 113
Semi-structures interview, 95, 96
Sequences
design, 27, 47
numbers, 113
process, 30
and series, 14
sets, 9
Seriation
and classification, 8
experiment, 140
sets sequence, 9
Series
APOS theory, 15
and sequences, 14
Set formation, 9
Set inclusion, 9
SETL. See Set Theoretic Language (SETL)SETL input, 11
!set random off, 60!set random on, 60Sets
action, mental Process, 68
axiom schema, 72
binary operation, 63–64, 71
and binary operation, 51
Cartesian product, 49
comparisons, 8, 83
concept, 35
construction, 184
contiguous intervals, 124
cosets, 177
create new sets, old ones, 68–69
description, 48–49
domain, 6, 118
finite and infinite, 182, 183
formation, 8, 9
funcs, 69–70functions, 30, 179
individual discovers, 112
inputs, 111
inv, 63ISETL, 60
mathematical notation, 60–61
mathematical objects, 13
natural numbers, 183
physical objects, 7
polynomials and matrices, 36
random off/on, 60
range, 6
scalar multiplication, 50
sequence, 9
single, 48
solution, 36, 37
spanning (see Spanning sets)
students, 74, 82
and tuples, 11
types, 35, 36
vectors, 35
Set schema, 67
Set Theoretic Language (SETL), 65
Shading
thought experiment, 145, 146
total square, 145
Sketch, 123, 127, 129–132
Slope, 114, 116, 118
Social component, 107
Solution, 141, 144, 147
Space, 111, 113, 122
Span
and spanning set, 33, 35–37
vectors, 54
Spanning sets
definition, 37
elements, 40
linear algebra concepts, 15, 111
and span, 33, 35–36, 38
Square, 145, 146
Stability of constructions, 108
Stage
APOS, 176
construction, 176
The Stage of concrete operations
APOS theory, 153, 164
elementary school, 151
Stages, 2–3, 17
inter-stage (see Inter-stage, schema)
intra-stage (see Intra-stage, schema)
relationship, class inclusion, 110
Index 251
Stages (cont.)schema, 114
“the triad”, 112, 113
trans-stage (see Trans-stage, schema)
Start, 138, 145–149
State at infinity (resultant state), 183, 184
Statements
construct, 30
declarative, 31
original, 32
positive integer, 32
return, 50
Static structure, 21, 25
Statistics
actions, 19–20
algebra and calculus, 18
encapsulation and objects, 21–22
interiorization and process, 20–21
Steps of interview analysis, 96
Strength of constructions, 108
Structural
cognitive, 114
conceptions, 15
geometrical, 113
learning process, 111
mathematical, 110, 128
mental, 124
problem-solving situation, 112
schemas, 110, 112, 118
Structures
cognitive, 52
and mechanisms, 27
mental, 27–29
Students’ thinking, 33, 114, 120
Subclass, 110
Sub-divisions, 140
Subgroups, 14, 101, 102
Subject
action conception, 144
APOS-based research, 138
data collection, 140
interviews and interpretations, 144
and object, 7, 143
particular mathematical problem
situation, 8
perform on them, 12
process, 149
process stage, 140
Subschemas, 110
Subspace, 36, 38, 54
SubString, 82, 83, 87–88
Subsystems, 110
Symbol, 9, 12, 15
Symmetries, 14
Syntax (as it relates to use of ISETL),
59–62
Synthesis, 113
Systems of linear equations, 15
T
Table of contents, 96
Tangent
circle, 53
line, 114, 116, 118
object, 53
Tasks, 34, 38, 45, 124
Teaching, 127
Teaching and learning groups
abstract algebra, 67
axiom schema, 68
binary operations (see Binary operations)
genetic decomposition, group schema, 67
group schema, 72–73
set formation (see Set)Teaching cycle, 47
Tennis ball problem
cardinality, 184
paradox, 184
Textbook analysis, 95, 103
Thematization, 2, 9, 13, 73, 175
APOS theory study, 129
calculus graphing problem, 129
condition and flexibly reassemble, 133
continuous function, 131–132
demonstration, coordinations, 130
function unique, 132
individual’s awareness, 128, 130–131
intervals, domain, 130
object, 131, 135
reflective abstraction, 128
remove, continuity condition, 133
and schemas, 24–25, 128, 135
Susan’s work, 132, 133
trans-property and trans-interval stage,
130, 131
work illustrating, 134
Theoretical analysis, 93–95, 182, 184
Theoretical perspective
APOS theory, 15
design and implementation, instruction, 15
Theories
APOS (see Action, Process, Object, andSchema (APOS) theory)
pedagogical, 47
set, 34
252 Index
Thought experiment, 145, 146
Tool, 111, 120, 125, 127
Topic, 110, 112, 122
Total entity, 15
Totality, 3–4, 76, 90, 91, 99, 131
decimal Process, 176
enumeration process, 183
process and object, 15
Traditional, 141
approach, 101, 105
instruction, 90, 91, 179
Trajectory, 29, 40
Trans-calculus, 112
Transcendent object, 140
Transcript, 144, 146, 148
Transcription of interviews, 97
Transformation, 103, 106, 138,
143, 149
analytic and projective geometry, 113
APOS interpretation, 186
dynamic, 30
graphing, 52
inter-stage, 114, 116
linear (see Linear transformations)
mathematical concept, 28
nonlinear, 41
object, 43
physical and mental, 175
reconstruction, 175
schema, 113
trans-function stage, 118
triad, 112
vector spaces, 42
Trans-function, 112, 118
Transition
action to process, 144
segments, 145, 147
Transition from one conception
to another, 108
Translation, 52
Trans-stage, schema
chain rule, 119–121
derivative, 118
functions, 118
interval, 125–128
mathematical definitions, 118
property, 124
The Triad, 112–114, 120
Triangulation, 95, 96
Trigonometric functions, 116
Tuples, 11, 60, 63, 64
n-Tuples, 35, 36Two variable functions, 34, 48
U
Unchanged, 133
Understanding
ACE teaching cycle, 105
APOS-based research, 96
genetic decomposition, 94
informal and formal, 45
inter-chain rule stage, 116
mathematical, 33, 34, 38, 128
researchers, 38, 109, 122
schemas, 110, 112, 122
statement, 32
students, 113, 131
variables, 36
Union, 27, 124, 125
Unique diagnostic tool, 27, 38
Unit
forming, 6
inclusion schema, 9
indistinguishable, 8
of measurement, 113
Universal quantifier, 43
V
Validity, 38
Validity of genetic decomposition, 99
Variable, 61–64, 83
binary operation, 48
concept, 36
expressions, 36
func, 50
solution set, 36
Variation of variations, 9
Variations, 10, 115, 116, 118
Vector problem
infinite zero vector, 184
natural numbers, 184
principle, accumulation, 184
Vectors
actions, 36
addition, 43
arbitrary, 35
linear combination, 39
pairs, 43
space (see Vector space)Vector space, 15, 111, 118
arbitrary vector, 35
concepts, 36
element, 36–37
genetic decomposition, 48–49
learning linear algebra, 48
in linear algebra, 21
Index 253
Vector space (cont.)linear transformation, 103
linear transformations, 41, 42
mathematical definition, 25
n-tuples and matrices, 25
process/object conceptions, 103
R2 and R3, 35
schema, 49–51
and students’ tendency, 35
subset, 36
Vertical, 126
View, 79–81
Vygotsky, L.S., 96
W
Walking on All Fours, 140
Ways of reasoning, 109
Written questions
action conception of cosets, 102
cosets, normality and quotient groups, 101
description, 100–101
encapsulation, 102
preliminary genetic decomposition, 101
Z
Zone of proximal development, 96
254 Index
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